Thoughts and experiences on various topics: puzzles, games, AI, collaboration, music, politics, and whatever else is on my mind

Alert readers will have noticed that my blog-posting schedule slipped last Sunday — I had promised to discuss the Power of Collaboration,  but it turned out that one of my current  collaborators was visiting from out of town.  So instead of writing about collaboration, I actively engaged in some fun and collaborative work!

I have been mostly self-employed for the last 15 years.  Working from home has distinct advantages, among them not having to commute, and having the freedom to set my own schedule.  The biggest downside I’ve encountered is a lack of collaborators to work with. My experience is that I’m many times more productive when I’m collaborating on some project, so for the last 5 years or so I’ve been actively seeking out collaborations.  I’ve been fortunate in the last 3 years to have found a number of excellent collaborators, and have been engaged in some really exciting projects:

1.  Monorail iPhone puzzle app  — a collaboration with my son Aaron during 2011.

2.  Android version of my Target Tiling (Tetris-inspired) puzzle app  — working with one of my brother’s former students.

3.  Working on a new iPhone puzzle app — as part of a team assembled by an MIT classmate of my son’s.

4.  Researching techniques for solving very large search problems — collaborating with my brother, Wayne, who is a CS professor at Westmont College.

The last 3 of these are actively ongoing, and I plan to write more about them in future posts.

The Power of Collaboration

It seems fairly obvious and natural (as a general rule) that 2 or more people working together can accomplish more than any one person working alone.  They can share the work and thus accomplish more.  But my experience is that the productivity of collaboration is not additive, but is in fact multiplicative.  When I’m working with a collaborator, I estimate that I am at minimum 10 times more productive  and sometimes even more than that!  My intent in this note is to examine how this works, and why it is the case.  I will be focusing on 2-person collaborations since that is what I have the most experience with.

Advantages of collaboration

1.  Dividing up the work

2.  Synergy of knowledge, skills, and ideas

3.  Maintaining focus and commitment

4.  Avoiding prolonged periods of being stuck on something

Dividing up work

If there is a fixed amount of work to do for a project, then 2 collaborators might be expected to accomplish it in half the time.  This assumes that both collaborators are equally capable of doing all of the work themselves with comparable efficiency.   If the collaborators are specialized so that each works best on separate sub-tasks, then dividing the work according to these specialties yields a greater than 2-fold increase in productivity.  In the most extreme case, imagine that neither collaborator has all the skills needed to do the project / task.  Then the time for either one working alone would be infinite!   But working together, if their skills together cover what’s needed, they can accomplish the task collaboratively in finite time.  That’s quite a speedup!

Synergy of knowledge, skills, and ideas

The knowledge and skills of even a 2-person team can be much greater than that of either individual alone.  Beyond dividing tasks according to respective expertise and specialization, the collaborators can share their knowledge and skills with each other so that each becomes individually more productive as well!  This seems very similar to the power of community learning environments (such as ESG at MIT), where the shared knowledge within the community enhances the knowledge of each individual community member.  There is also synergy of ideas — collaborative brainstorming can be both fun and productive, with each collaborator contributing ideas, and feeding off of (reacting to) the ideas of the other.  When it comes to evaluating ideas, having 2 points of view can be beneficial as well.

Maintaining focus and commitment

I have a very great tendency to get distracted (I’m probably ADD or something along those lines).   When I’m actively collaborating (in realtime) with someone, I find that I’m much more able to maintain focus and “stay on track”.   Even when not in realtime, it is still helpful to have a commitment to one’s collaborator — putting off or avoiding things is letting your partner down.

Getting unstuck more quickly (avoiding staying stuck for prolonged periods).

A very specific and powerful aspect of collaboration is reducing the length of stuck times!   When working on a task alone, I often run up against obstacles which stump me (sometimes for prolonged periods).  This can occur for many reasons, including lack of a specific skill or piece of knowledge, or simply not being able to immediately solve some challenge.  Very often, one’s collaborator can offer a key insight or suggest a useful tool or mechanism that gets me quickly unstuck.  Without that outside help, I am much more likely to stay stuck for an extended period.  Similarly, my collaborators can often “rescue” me from pursuing a dead end, by helping me recognize earlier on that it’s either hopeless or unproductive.

Some slight disadvantages of collaboration

There are also a few small disadvantages of collaboration, for example:

1. Overhead in scheduling / finding time to work together

2. Having to compromise (sometimes involving lengthy discussions to resolve differences)

But overall I find that the powerful benefits of collaboration far outweigh any disadvantages!

Next week:

Collaboration “case studies” — exploring more concretely some of my experiences in working with collaborators.


Today I’ll wrap up the current discussion of Target Tiling with a look at how to achieve symmetry.  Also, I’ll describe something I call pseudo-symmetry,  which is similar to and behaves somewhat like symmetry.

Achieving Symmetry

For Target Tiling on a board of width = 5, there are a total of 32 different row patterns (5-cell patterns of filled and empty cells).  Of these, there are 8 that are left-right mirror symmetric (they are the same when order of cells is reversed).   Last week I illustrated the 6 “non-trivial” symmetric patterns (omitting both the empty pattern and the completely filled pattern).  That leaves 24 row patterns that are non-symmetric.    These non-symmetric patterns match up into groups of 4 related patterns:  each group containing the 2 mirror reflections of one non-symmetric pattern, each occurring twice — once with the center cell filled and once with the center cell empty.  So fundamentally there are 6 different asymmetric patterns (each generating a related group of 4 out of the 24 patterns).

The next image illustrates all 6 possible non-symmetric row patterns (with center-cell empty, and omitting the mirror reflections):

Canonical asymmetric rows (with center cell empty)

Canonical asymmetric rows (with center cell empty)

Each of these asymmetric patterns (and the variants with reflection and center filled) can be first transformed into a symmetric pattern, and then cleared by a symmetric wipeout using the symmetric macro-operators described in my blog post last week.

I will present an illustration  using a single 1-cell pattern, starting with just a single cell in the 2nd row (1 cell left of center).  Here is the starting board position:

Start pattern for example of achieving symmetry

Start pattern for example of achieving symmetry

This non-symmetric pattern can be transformed into a symmetric pattern with just 2 T-piece placements. The first T piece is placed to the right with its stem pointing right, which yields this position:

After first T-piece is placed at right

After first T-piece is placed at right

Now place the second T piece to the left with stem pointing up, yielding this symmetric position (before the filled row clears):

Symmetry achieved (before row clear)

Symmetry achieved (before row clear)

After the filled row clears, the pattern is still symmetric:

Symmetry achieved (after row clear)

Symmetry achieved (after row clear)

This symmetric pattern can be wiped out with the application of symmetric macro-operators as discussed last week.

The macro-level description could be:

Repeat 2 times:  DO  [ Clear-Center-Hole    +   Clear-3  ]

This takes a total of 14 moves.

Transforming Pseudo-symmetry into Symmetry

Here is a second example, of clearing a non-symmetric pattern by first transforming it into a pseudo-symmetric pattern (symmetric if rows were superimposed on one another).

To illustrate this, consider the challenge of clearing a T-shaped start pattern (where the T is off-center to begin with):

Start with off-center T at right

Start with off-center T at right

This position is non-symmetric.  There is an easy pattern I discovered for clearing it, though.  The first move is placing a T on the left with stem down, achieving a nearly symmetric position (what I call pseudo-symmetry):

Pseudo-symmetric pattern after placing T at left

Pseudo-symmetric pattern after placing T at left

The only feature that breaks symmetry here is that the single filled cells in the left and right columns are in different rows.  Ignoring this difference, the position is symmetric.  It turns out that this can be transformed into an actually symmetric position with 2 moves.  The first move is to place a T at far right with stem in so as to fill the hole at the right of the 2nd row.

After filling hole at right (but before row clear)

After filling hole at right (but before row clear)

When the row clears we arrive at this pattern:

After row clear

After row clear

Now a 2nd T piece placed at the left (with stem pointing right) yields a true symmetric position:

Symmetric position after T placed to fill left hole

Symmetric position after T placed to fill left hole

This symmetric position can be wiped out with just 1 more piece filling the T-shaped hole.

There are many more patterns to learn, but with practice, one builds up skills for clearing all the non-symmetric patterns.

My experience is that symmetric patterns are significantly easier to clear, so it is worthwhile to learn how to achieve symmetry from non-symmetric patterns.

Check out Target Tiling — it’s free, it’s fun, and it’s fantastically challenging!

Instructions on how to download and run my game are on my web page:

Next Week:  The Power of Collaboration

Using symmetry to solve width 5 Target Tiling puzzles using the T-piece

By symmetric positions, I mean board patterns that have left-right (reflective) symmetry, that is, they are unchanged if the order of the columns is reversed.  In a symmetric board position, every individual row will itself be symmetric.   But of course not all patterns will be symmetric, so for those non-symmetric patterns, the challenge is to achieve symmetry.

Why is symmetry useful in the first place?

It would be pointless to achieve symmetry unless there were something to be gained by it.  In fact there is a lot to be gained:

1.  There are far fewer symmetric patterns than non-symmetric ones.

2.  It is easier to recognize familiar symmetric patterns.

3.  In solving symmetric positions, it is possible to do so while maintaining symmetry!

4.  The number of “operators” that maintain symmetry is much smaller than the set of all available operators, so there are fewer choices of what to do at each point (in search terms, the branching factor is smaller).

Exploiting symmetry

Taking advantage of symmetry has two parts:  achieving symmetry,  and then maintaining symmetry while simplifying and solving.  I will discuss each of these after presenting some examples.

Examples of symmetry

Here is one example of a symmetric position with 6 partially filled rows:

Symmetric board illustrating all 6 possible symmetric rows (except filled and empty)

Symmetric board illustrating all 6 possible symmetric rows (except filled and empty)

In fact, every possible symmetric row pattern appears in this diagram.  There are 6 such symmetric row patterns.  Notice that rows 3 and 5 (counting from the bottom) are the cases I discussed last week:  Center-hole, and 3-in-the-middle, respectively.   I explained how each of these patterns could be cleared by an appropriate macro-operator:  the Clear-3 macro for the 3-in-the-middle pattern, and the Clear-Center-Hole macro for the Center-hole pattern.  It is possible to learn a “clearing macro” for each of the other 4 row patterns, as well.

Solving While Maintaining Symmetry

The key to maintaining symmetry is to only place T-pieces so that the result is still symmetric. There are 2 cases I find useful:

1.  Placing a single T-piece in a symmetric position

This can only be done in 2 ways, since there are exactly 2 rotations of the T-piece that are left-right symmetric, and the piece must be placed exactly in the center (straddling the center the column).

Symmetric placement of T-piece:  in center with stem-down

Symmetric placement of T-piece: in center with stem-down

Symmetric placement of T-piece: in center with stem up

Symmetric placement of T-piece: in center with stem up

2.  Place two T-pieces so that together they form a symmetric pattern

Again, there are only 2 ways to do this.  The pieces cannot cover any of the center column, since they would both have to cover the same cell, which is not possible. Here are the 2 placement patterns:

2 Symmetric T-pieces, stems facing in

Placing 2 T-pieces symmetrically, stems facing in

Placing 2 T-pieces symmetrically, stems facing out

Placing 2 T-pieces symmetrically, stems facing out

Either of these 2-piece macro-operators will maintain symmetry when they are applied to any symmetric position. Taken together, this set of 4 operators (2 single-piece + 2 double-piece) are sufficient to solve (i.e. Wipe Out) any symmetric starting position. Let’s give names to these operators:  T1-Stem-Down,  T1-Stem-Up, T2-Stems-In, and T2-Stems-Out, respectively. Looking back to the examples from last week, we can see how the macros Clear-3  and  Clear-Center-Hole  are actually composed from these 4 operators:

     Clear-3   =   T2-Stems-In  +  T1-Stem-Down


    Clear-Center-Hole  =  T1-Stem-Down  +  Clear-3

where “+” means “followed by”.  Note that any macro-operator composed from symmetric operators will itself be symmetric!

Solving the Illustrative Symmetric pattern we started out with:

Symmetric board illustrating all 6 possible symmetric rows (except filled and empty)

Symmetric board with all 6 possible symmetric rows

We start by applying T2-Stems-In to arrive at:

Result of applying  T2-stems-in operator

Result of applying T2-stems-in operator

Next it is natural to apply T1-stem-down (since there is a “T-shaped hole” in the middle of the top 2 rows):

Result after T1-stem-down

Result after T1-stem-down

Now, the top row is a “3-in-the-middle” which can be cleared by the Clear-3 macro, yielding:

After Clear-3 macro operator

After Clear-3 macro operator

Here the top row is a “Center-Hole” pattern, which can be cleared by the Clear-Center-Hole operator to leave:

AFter Clear-Center-Hole macro

AFter Clear-Center-Hole macro

Dropping a T-piece with stem up into the middle of this pattern transforms it into:

After T1-Stem-Up

After T1-Stem-Up

Filling the 2 holes with 2 T-pieces using T2-Stems-Out, reduces the pattern to 2 cells in just a single row:

After T2-Stems-Out

After T2-Stems-Out

Almost there!  A simple T1-Stem-Down fills the center hole, and yields 2 rows of “3-in-the-middle”:

After T1-Stem-Down

After T1-Stem-Down

The final two rows, each being examples of “3-in-the-middle”,  can be cleared by two successive applications of Clear-3.

That solves the puzzle!!   Notice how the positions remained symmetrical at each stage of the solution.  Typically rows were cleared one by one.

To be continued …

Next time:  Achieving Symmetry (transforming non-symmetric patterns into symmetric ones)

Last week I discussed an extremely advanced Target Tiling challenge.  This week I’d like to introduce and discuss some simpler strategies for solving basic Target Tiling wipeout challenges.  The two key ideas are:

1.  Learning Macro-patterns

2.  Achieving and exploiting symmetry

Each of these techniques helps simplify the solving of wipeout challenges.

Discovering and using macro-operators

You may recall from previous posts that a macro-operator is a fixed sequence of simpler operations that achieves a specific transformation.  I’ll illustrate this with something I call the Clear-3 macro.  It applies in the WipeOut Challenge using the T-piece on a board of width 5.   Here is the starting board state for the macro:

Start of Clear-3 Macro

Start of Clear-3 Macro

It is natural to try filling the right and left holes as follows:

Left hole filled, about to fill right hole

Left hole filled, about to fill right hole

Then, after the filled bottom row is cleared, we arrive at this position:

[Note that for today’s discussion, we are using the original (single row) version of the Row-Clear Rule]

WipeOut coming with next move!

WipeOut coming with next move!

This is just 1 move from a WipeOut  (clear board).  The last piece (the 3rd in the macro sequence) clears the remaining 2 rows:


Note that the difference between the start and the final positions is the removal of the 3 filled cells in the middle, hence the name Clear-3.

The Power of Macros

The Clear-3 macro allows us to clear any row of 3 filled cells situated in the middle of the width 5 board, using T-Pieces.  As an example, suppose we have a rectangle formed by 4 rows of “3 filled cells in the middle” of each row, as in the next diagram:

4 Rows of "3 in the Middle"

4 Rows of “3 in the Middle”

This pattern can be easily wiped out in 12 moves, simply by applying the Clear-3 macro  4 times in succession, clearing each row in turn.  Although the solution involves 12 piece placements,  conceptually it is very simple:

    Repeat 4 times:  Do Clear-3

The fundamental power of macros is that they allow planning at a more abstract level (hiding or ignoring low-level steps in the planning / problem-solving process)!

Using Clear-3 to solve a new problem

What if we are faced with this next position, where a row is filled except for a “hole in the middle”:

Row with 1 hole in middle

Row with 1 hole in middle

A single move (placing a T-piece stem-down into the center hole) yields the familiar “3 in the Middle” position:

Row of 3 in the Middle

Row of 3 in the Middle

This can be eliminated (WipedOut) with a single application of Clear-3.  Note that the sequence involves 4 Piece placements, but conceptually is just 2 steps (fill hole  +  Clear-3).

This 4-move sequence is a new macro, allowing us to clear any row filled except for a hole in the center.  Let’s call this macro Clear-Center-Hole.

Clear-Center-Hole can be applied iteratively, as well.  Suppose the board is filled 4 deep with a column of holes running down the center:

4 Rows of "Center Hole"

4 Rows of “Center Hole”

This starting position can be solved by 4 applications of Clear-Center-Hole.

The entire sequence takes 16 moves, but conceptually it is simple and elegant:

     Repeat 4 times:  Do Clear-Center-Hole

These macros, by themselves, do not suffice to solve (WipeOut) all positions – there are many other patterns, most of which are not even symmetric.  Skill and mastery involves learning patterns for transforming given positions into other simpler / known positions (such as the 3-in-middle and center-hole above).   One technique I’ve found quite useful is to transform positions so as to make them symmetric.  Once symmetry is achieved,  moves can be chosen so as to maintain symmetry, which greatly reduces the number of patterns one is faced with.

To be continued …

   Next time:  Achieving and Exploiting Symmetry in Target-Tiling positions

Continuing from last week: more on my Target Tiling puzzle game (inspired by Tetris), including a novel extension to the Tetris row-clearing rule, a seemingly impossible challenge, and a very surprising solution!

Modifying the Row-Clearing Rule

Recall that in Tetris (which inspired my Target Tiling puzzle game), there is a row clear rule:

After each piece placement, any rows that are completely filled are eliminated,  and the rows above move down to fill the vacant space.

When I had become rather expert at all the earlier variations (varying widths, piece types, piece sequences, and targets), I began thinking about new ways to challenge myself.  I thought about the row clearing rule, and asked myself:

What if it wasn’t enough to fill a single row to get a clear?  What if you had to fill 2 consecutive (adjacent) rows before they would clear? What if the threshold was 3 rows?  or 4 rows?   This seemed like an interesting challenge, and also related to the notion of Tiling (with pieces)  which conventionally involves completely filling a given space with available pieces.   So I implemented this variation of row-clearing in my simulator, generalizing the row-clear rule to an  N-row-clear rule (where N could be 1, 2, 3, or 4):

When N (or more) consecutive rows are completely filled, they are cleared as with the original row-clear rule, with any higher rows moving down to fill the emptied space.

NOTE that when N=1, this rule is exactly the original Tetris  rule for row clearing.

It turns out that the variants of row-clearing with N= 2, 3, and 4 are interesting and present some quite complex challenges.  I started playing with N=2 (Double variant), and learned new patterns for solving the various width, piece, and target patterns.   When I felt proficient with that, I advanced to N=3 (Triple), and finally N=4(Quadruple).

A Nearly Impossible Challenge

When I tried the N=4 (Quad) variant of Width=5 using T-pieces, I thought I may have reached the limit of solvability.  I was stumped for quite a long time, and was close to giving up, declaring it “too hard, and maybe impossible”.   Then, one day, I tried something desperate – a totally counter-intuitive Tetris piece placement,  and low and behold, this enabled me (with long macro-sequences) to actually solve it!

Here is an example position illustrating what was stumping me for the longest while:

This stumped me for a long time!

This stumped me for a long time!

Everything I tried seemed to lead nowhere.  I was about to give up, when I discovered this extremely counter-intuitive T-piece placement.  This discovery led eventually to a solution to this and other Width=5 T-piece Quad puzzles!

A Surprising Counter-Intuitive Discovery!

A Surprising Counter-Intuitive Discovery!

This placement is counter-intuitive because it violates the natural Tetris heuristic:

      Try not to create any covered holes!  

This piece-placement violates that heuristic by creating 2 covered holes!  On first examination, it looks like it fails to get closer to filling any rows.  The surprise was that the top row is filled except for the hole in the 2nd position, and this top row can be eliminated (by first achieving symmetry and then wiping out everything above and including this row (without modifying anything in the pattern below).

It takes 9 moves to clear the top row, and another 13 to wipeout the rest, for a total solution in 23 moves (counting the “surprise” move).  Here are some snapshots to illustrate this solution sequence:

After 5 more moves

After 5 more moves

Is this progress?  Look what happens after just 4 additional moves:

Top Row Cleared!

Top Row Cleared!

The Top row (created by the “surprise move”) is now cleared.  This is finally looking like progress!

Down to 1 Row - after 4 more moves!

Down to 1 Row – after 4 more moves!

An additional 4 moves lead to this position, just 1 move from a solved WipeOut:

1 more move to solve!

After 4 more moves – just 1 more move needed to solve!

Placing the last T-piece in the hole will clear the final 4 rows, leading to an WipeOut (empty board)!

This solution took a total of 23 piece placements starting from the original Stuck Position. 

I was incredibly excited by this discovery.  This “key (surprise) move” insight, and the patterns learned to clear from it, enabled me to solve the entire class of T-piece Width=5 Quad puzzle patterns.  What I mean by solving the entire class is being able to solve (WipeOut) any initial pattern given enough clear board space to work with at the top of the board.  I could also solve the related class of Target problems.  What seemed impossible at first, was actually quite solvable, but required a very specific (surprising and non-intuitive) insight!

Next week:  discussion of some more basic Target Tiling solving strategies (macro-patterns and symmetry)

Thinking Deeply About Logic Puzzles 

If you know anything about me, you know that I’m passionate about puzzles.  I particularly like logic puzzles, by which I mean puzzles that require logical thinking to solve.  In these notes I will try to “think deeply” about puzzles: What is a puzzle?  What makes a good puzzle?  How do I create new puzzles?

What is a puzzle?

There are many different types of puzzles, but the primary common feature is that they  have a solution!  Some puzzles explicitly define the elements of the puzzle and the valid operations or moves.  Other puzzles don’t even make clear exactly what the goal is, and part of the puzzle is to “figure out just what the puzzle is”.  Still other puzzles have hidden elements that may get revealed (e.g. MasterMind, or puzzle boxes with hidden mechanisms).  Math problems and scientific research projects can also be viewed as kinds of puzzles, some explicit, others very open-ended.  There are jigsaw puzzles where one has to use piece shapes and visual clues to assemble a target picture.  There are popular paper-and-pencil puzzles (like KenKen and Sudoku).  With the development of computers and videogame systems, there are now a wide range of interactive puzzles (and video games).

I personally am particularly interested in puzzles which are explicitly stated (goal is clearly defined, and the elements and operators are well-defined), and can be solved by logical thinking.  I don’t particularly care for puzzles with hidden goals or hidden elements.  Nevertheless, the class of explicit puzzles is quite broad .   I enjoy physical puzzles, paper-and-pencil puzzles, and interactive (computer and smartphone) puzzles, among others.

A Few of my Favorite Puzzles

1. Physical puzzles:   Rubik’s Cube,  Hi-Q Peg Solitaire, Hi-Q Dominos, Rush Hour, numerous other sliding-block-puzzles  (and many more!)

2. Paper and Pencil puzzles (newspapers, magazines, books):  KenKen,  Sudoku, RoundTrip and OneWayTrip (Monorail type path puzzles)

3. Computer and videogame puzzles:  Tetris,  Target Tiling (my Tetris-inspired puzzle game), many of the Soleau Software puzzles, plus my Grand Tour (Monorail/RoundTrip), and One-Way Trip applets.

4. SmartPhone puzzles:  here is a link to My Favorite Interactive Logic Puzzles.  I especially love Vubu, Relix, BoxedIn, Blockhouse, and Monorail, but all the puzzles listed are great (imho)!

What Do I like about these puzzles?

a.  They are challenging and fun to solve

b.  Most involve visual / geometric elements

c.  I can develop skills and expertise at solving  (a variety of reasoning strategies that can be learned!).  Best of all if there is a hierarchy of skills!

d. They are reasonably time-limited (typically solvable in hours — not days, weeks, or years)

Creating a new puzzle – case study of Target Tiling (inspired by Tetris)

I became a Tetris addict when it appeared back in the late 1980’s, playing first on the Mac SE, later on the NES (Nintendo Entertainment System).  I loved the geometry, the challenge, and even the music (Russian folk tunes)!   I was inspired to write my own Tetris simulator, so I could experiment with variations of the game.  Over time, this project grew into a rather rich and extremely challenging interactive puzzle game (my Target Tiling – available for free download from my web site)

1. Identifying the elements:

Board size (Tetris had a fixed 10-wide board),

Pieces (Tetris had 7 pieces, each covering 4 grid cells) that could be rotated and placed in position

Row-clearing rule (completely filled rows are eliminated and the cells above move down to fill the vacated space)

2. Playing with (varying) the elements

Once I had written a general-purpose Tetris simulator, I began varying and playing with the elements.   I tried different board sizes, both narrower (4, 5, 6, 7) and wider up to 20 or 30.  I introduced different piece shapes,  again, both smaller (1, 2, and 3-cell pieces), and larger (the full set of Pentomino shapes, which cover 5-cells each).  I also varied the speed at which pieces would fall on their own, but I found I much preferred slower speeds (I also included a “Pause” feature”) – in general I’m not a big fan of timed puzzle solving or games – I prefer to think in a more relaxed mode.

3.  Varying the puzzle objective

Tetris is fundamentally a game of survival — typically the pieces speed up until they fall so fast that you can’t control them, at which point the board “fills up” and you lose.   I much prefer puzzles and games that I can “win” (solve).   At some point I latched onto the challenge of “completely clearing the board”.   I thought this made a nice clear objective, and provided satisfying closure when achieved!   I first tried this (as a self-defined objective) in Tetris on the NES, but it was extremely difficult to achieve because the “Next Piece” was always chosen randomly – so one couldn’t plan ahead very far.   I’d try to achieve a board pattern with the “bottom rows” filled except for a “hole” in the shape of one of the pieces.   Even doing that is quite difficult, but once accomplished one has only a 1-in-7 chance of getting the necessary shape as the next piece.  It struck me that this “puzzle objective” (clearing the board) would be much more interesting if the next piece was chosen deterministically, as this would allow planning.

4.  Simplification  (thinking deeply about simple things)

So I experimented with various simplifications / modifications

a.  Make the board width smaller (5, 6, and 7, seemed to work well)

b.  Use just a single piece type (this was the simplest way to make the next piece choice deterministic – you always get the same piece shape!)  Through experimentation I found that the T-piece and the L-piece (or J-piece) seemed the most interesting.  The Z-piece was impossible (on a 2D board), and the 2×2 square and the 1×4 “I”-piece seemed rather boring.

5. Skill learning

Using the T and L pieces on boards of width 5, 6, and 7 turned out to be very challenging and interesting.   Over time, I was able to develop skills that let me reliably and fairly quickly solve these puzzles.   I used a random start position (formed by dropping a specified number of pieces of the given type), and would then proceed to work the pattern down (reducing the number of rows) and finally clear the final bottom rows.  I found this both challenging and immensely enjoyable.   I particularly like the set of skills I developed that made me an “expert” at these puzzles!  These skills formed a natural hierarchy, with higher-level skills building on and using lower-level skills.  At the lowest levels there are standard Tetris skills (using the controls to rotate and place pieces in desired target locations, and exposing and filling “holes” to reduce the height of the remaining cell pattern).   The final “WipeOut” skill (clearing the board exactly) involved learning to recognize specific “solvable patterns” and applying the associated move-sequences (macro moves) to solve them.  Some of these macro-moves simply transformed one recognized pattern into another (simpler) pattern that could then be solved by subsequent macros.

6. Adding incremental complexity

After I became skilled at these basic puzzles, I started thinking about adding new challenges.

a. Multiple pieces

I tried a variation using both the T and L pieces (alternating so that the piece sequence was still deterministic so as to allow planning).  Having 3 or more pieces can be extremely challenging, even when still deterministic, but experimenting with those possibilities led to a virtually endless set of rich challenges.  I even tried returning to the complete set of 7 original Tetris pieces (presented in a deterministic fixed / cylic order) – that was difficult but still a fun challenge, though I still preferred narrower boards — width 10 gets rather tedious and takes a really long time to solve.  I found that widths 5, 6, and 7 were most enjoyable.

b.  Target Pattern (generalization of WipeOut)

At some point it occurred to me that I could generalize the puzzle objective from WipeOut to achieving a specified Target pattern (a particular set of marked cells to be filled).  The WipeOut objective could be viewed as the special case of the “Empty Target pattern”.   Initially, I would specify some cells to be left filled in the bottom row.  This led to a new level of skill based on doing a “wipeout” in “mid-air”.  Let me explain.   If the target was a single cell in the bottom row, for example, then one could begin by doing a full WipeOut (clearing the board), but this would not “solve” the target puzzle, since the target cell is empty, and needs to be filled.   Next, one could place a piece so that it covered the target cell, but no other cells in the bottom row – this might involve “standing the piece on end”, for example.  This fills the target cell, but still doesn’t solve the puzzle — the objective is to match the exact target pattern by filling all the marked target cells and no other cells!).   In our example, this can be achieved by wiping out rows 2 and above, while being careful never to fill any of the empty cells in the bottom row — I call this a mid-air wipeout.  It turns out to require concave pieces (such as the T and L) which can then “hang over the side” of filled cells, so as to fill more and more of the rows to be cleared (wiped out).  This introduced an entire new level of skills and patterns!

c.  Mid-air Targets

There was an “Aha!” experience I had while playing Tetris on the NES.   I asked myself it it might be possible to “suspend a piece in mid-air”, by which I mean have piece “floating” without obvious support (either from below, or cantilevered from a filled cell to its right or left).   At first blush, it seemed like there was a kind of “gravity” at work in Tetris, so maybe this was impossible.  But Tetris-gravity is different — when each row clears, cells above that row only move down 1 cell, in particular, they do not keep falling!   So I set myself the challenge of “floating a piece”, and soon discovered that I could create a mid-air platform (2 or 3 nearly filled rows with a piece-shaped hole) cantilevered from a single support cell (that is the row below has 1 cell filled but the rest of that support row is empty).

Here’s an attempt to illustrate:

This graphic shows a platform (rows 2 and 3, not counting the red base row) with a T-shaped hole.  A T-piece is descending in columns 3-5, and will “fill the hole” in the platform:


After the piece fills the platform hole, rows 2 & 3 will be filled and so will be removed, and the 4 cells resting on the platform will move down 2 spaces, ending up as “floaters” in rows 2-4.  The floating T-piece will not fall further, it is statically floating:


 d.  3-D Geometry

You might notice that the pieces are drawn in a “sort-of” 3D perspective.  In fact, the current version of Target Tiling supports both 3D board grids and 3D pieces.  Check out the game on my web page to try it out:   Target Tiling  (this gets you a page with instructions on how to download and install – it’s all free!)  Eventually I hope to release a version for smartphones, but the current puzzle game runs on MacOSX, Windows, and Linux.  To run it, you need to download the LispWorks PE  (which is also free) – just follow instructions on my page.  If you do try it out, please send me your reactions and thoughts. Thanks!

… more next week

Musical Notes

[Apologies for the “blogslip” this week  (missed my Sunday target)…]

Did I mention that I love music?

I love to listen to music, play music, and have tried my hand at composing/songwriting.

Listening to music

I love to listen to music.  When I’m working on my computer, I love to have iTunes playing in the background.  My favorite genre is blues, but I also enjoy classic and progressive rock, as well as newer groups such as ColdPlay and Muse.  I have a long list of “favorite blues musicians”, but one that I particularly think deserves to be better known is Sherman Robertson, so I’m giving him a special mention.

When I can afford it, I enjoy going to blues clubs, concerts, or outdoor blues festivals.  This August I enjoyed both the Gloucester Blues Festival, and the annual Blues & Brews Fest.  In June I was able to attend my first Chicago Blues Festival (in Grant Park).  The highlight for me was the pre-Festival concert at Buddy Guy’s Legends, where I heard Shemekia Copeland and Kinsey Report (both great!). Also first heard Quinn Sullivan – a young teenage guitar prodigy who is being mentored by Buddy Guy!

I also listen to music on the radio – though the ads get to be extremely annoying.  I regularly listen to Carter Alan’s Sunday Morning Blues on WZLX FM Boston.  My other regular show is Blues Breakers with Tom Marker on WXRT FM Chicago – I listen to this via the Internet.   I regularly record both these radio shows using the service of, a free service that usually does a decent job of recording radio shows available on the internet ( also has a paid service that let’s you record more shows and use more storage to keep recordings longer).  I listen to Sunday Morning Blues “live” on Sunday mornings (9am-noon) when I wake up in time (which I haven’t done consistently of late).  I then listen a 2nd time (or 1st time if I overslept) with the recordings after the playlist is posted (usually by Monday or Tuesday).

Playing Music

I love to play music, too!  I started piano lessons around 1st grade (thanks to my father who was my piano teacher!)  Piano / keyboards is still my favorite instrument.  I also studied trumpet from 4th to 12th grades, and even tried Jazz Band at MIT, but I never loved trumpet like I do piano — the piano is just so much more versatile!   I got my first electronic keyboard as a grad student – an RMI that I played through a Fender Reverb amp.  I had fantasies of joining a rock band, and even tried the occasional audition, but never got off the ground with that.   I tried to learn guitar, but never felt comfortable with it — too hard for me to memorize all those different chord patterns.  I also taught myself recorder, which I found quite enjoyable.  I especially enjoy the Renaissance piano/recorder duets I have, and always am pleased when I have someone to play with (I’m to do either the piano or recorder parts!).  My favorite classical music for piano are Beethoven sonatas, especially my life-long favorite, the Pathetique!  While working at GTE, I enjoyed collaborating with a flute-playing colleague to play the Bolling Suite for Jazz Flute and Piano – a wonderful piece of music.

Song Writing and Improvisation

I enjoy piano improvisation, and have also tried my hand at some song writing.  I wrote a couple songs back in high school, then some more in college.  In the aftermath of my divorce, I channeled my emotions into writing a fair number of songs.  I even audited a song-writing seminar at MIT.   In that seminar, we had to write a new song each week – at first I thought that would be impossible, but it turned out to be fine.   The deadline of “finishing” a song in a week meant that I couldn’t obsess endlessly trying to improve or perfect it.   There is a counterintuitive interplay between “quantity” and “quality”.  Surprisingly, striving for quantity (lots of songs) can also lead to better quality (compared to tinkering with a single song trying to perfect it).  I’m reminded of the writing seminar I participated in at ESG at MIT – we consciously tried to write continuously (stream of consciousness) without censorship.  This ended up being very liberating.   I also recall reading an anecdote (I forget where) about an art teacher who divided his ceramics class students into two groups:  the first to aim for quantity, the second for quality.  The surprising outcome was that those students striving for (and being judged by) quantity, also ended up producing higher quality work!  My “mini-theory” to explain this is that the quantity group is getting more experience by trying more new and different things, and thus also encounters more learning opportunities.

Open Mic nights

This Spring, I dusted off some of my old songs, and started performing them at Open Mic nights in the area.  The Open Mic at Nourish Restaurant in Lexington Center turns out to be very convenient — I only live a block away, and there is already an electronic keyboard there, so I don’t have to schlepp my own!   For better or worse (and often worse in my case), the performances there are recorded, so I now have audio available of many of the songs I’ve performed.

This Summer, I had the great pleasure of collaborating with my daughter Rachel (who is a violinist – music conservatory major).  We performed arrangements of several of my songs, plus a cover of a Bonnie Raitt song.  I really enjoyed this collaboration.  I was reminded of how much I value collaboration in all aspects of life – not just technical development work.

I finally collected the Nourish recordings to make available (in all there crudeness and roughness) on my web page.  There is now a link to my “Music page” from my “Home page”:

The direct link is:

There are even two videos of me and Rachel (kindly recorded and shared by another Nourish open mic participant).

Caveat:  These are not polished performances, so don’t expect a lot.  I hope, nevertheless that you can derive some enjoyment from my sharing them.

“If music  be the food of love, play on …”  — William Shakespeare

There are many books that have influenced and shaped both who I am and the directions my life has taken.  I can’t mention them all, but I’d like to note a few highlights.

Robots and Artificial Intelligence

The first robot book I remember was Isaac Asimov’s  I Robot,  a collection of science fiction short stories about robots, especially the “Robbie” story.  These stories introduced me to the idea of “intelligent robots”, and had enormous impact on my career choice to pursue studies and work in Artificial Intelligence and Machine Learning.   It inspired me to commit much of my life to what I consider one of the most challenging and fascinating intellectual questions:  What is intelligence, and How does it work.  Studying human intelligence is just part of that quest – a deeper understanding, I believe, is to be gained by discovering the fundamental principles of intelligence (in all its manifestations), and applying that understanding in creating new forms of intelligence, significantly in machines such as computers and robots.

Another brilliant book is Mindstorms by Seymour Papert, my advisor in grad school at MIT.  I highly recommend it to anyone with interests in learning and education, AI, and understanding thinking.   There are many important ideas in this book, but the single one that most deeply influenced me was Papert’s analogy between “artificial intelligence” and “artificial flight”.  He pointed out that studying neurons in the brain in order to understand intelligence, was rather like studying feathers in order to understand flight.  Early attempts at engineering flight tried to mimic bird flight, with mechanical flapping wings.  In hindsight we realized that the fundamental principles of flight involve aerodynamics and not flapping motion or feathers!   The conclusion suggested is that machine intelligence will grow out of a deeper understanding of higher-level principles governing thinking, and not the low-level substrates such as neurons (or electronic gates, for that matter).   This argument convinced me to abandon studying neurophysiology to gain insights into how the the brain (and intelligence) worked.

Douglas Hofstadter’s book Godel, Escher, Bach was another tour-de-force.  He brilliantly weaves together  themes from mathematics, logic, and computation, with Escher’s graphic art, and Bach’s music.  The book is an exploration of self-reference and formal systems.  Truly fascinating!   It had enormous appeal to me, given my diverse interests in math and computation, my passion for music, and  my enjoyment of Escher’s etchings, especially their exploration of visual-spatial ambiguity and self-reference.

Puzzle Books

I have already mentioned how deeply I was influenced by the writings of Martin Gardner, including his many collections of recreational math puzzles and essays.  Another of my favorite “logic puzzle” books is Raymond Smullyan’s brilliant Chess Mysteries of Sherlock Holmes.   It is a book of chess puzzles, but not traditional “mate in 2 or 3 move” puzzles, but rather retrograde analysis puzzles,  where one has to reason backwards from a given position.  Typical questions posed are “What was the last move?” or “Is it legal to castle in this position?”.  The reasoning and logic of these puzzles is incredibly rich and deep, and naturally appealed to me both as a puzzle-lover and a chess-player.  Smullyan brilliantly presents these puzzles in the context of stories about Sherlock Holmes and Dr. Watson, with occasional references to Moriarty thrown in!  A truly whimsical and thoroughly enjoyable presentation of a delightful collection of retrograde chess logic puzzles!

I also loved the Winning Ways volumes by Berlekamp, Conway, and Guy.  I’m totally in awe of the amazing recreational mathematics collected in these volumes!  Deep analysis of many fascinating games and puzzles.  I thought I had become fairly expert at “peg-jump solitaire” puzzles, with my discovery of “macro-operators”.  but I was humbled to discover the depth and breadth of the analysis covered in Winning Ways — yes, they had “macros”, which they called “packages”, but they went so much further in developing mathematical tools for analyzing questions that I hadn’t even thought to pose, as well as solvability of classes of problems.  I don’t know if I’ll ever finish reading all of the collected puzzle and mathematical wisdom in Winning Ways, but it is an amazing menu of “food for thought”!

Spiritual Books

My spiritual journey has been long and winding.  I plan to devote one or more future blog posts to recounting this journey in more detail.  There are a number of books that have greatly influenced my spiritual seeking and thinking.  Will Durant’s The Story of Philosophy  started me thinking about many fascinating questions, including “Is there a God?”.  After reading about the thinking of many philosophers on this issue, I came to the conclusion that there was no way to “prove the existence of a God”.   I particularly esteemed the philosophical approach called Empiricism, which reflected my scientific world-view.  I couldn’t see how any experiments could bear on the question of God or the supernatural, so I concluded that the simplest explanation was that religions and the notion of a “God” were based on myths and fables, with no bearing on “reality”.

In high school and college, I came across Edgar Cayce, famously known as “The Sleeping Prophet”.  The book There is a River by Thomas Sugrue is a fascinating biography of Edgar Cayce.  This book, in particular (which I highly recommend to anyone), opened my mind to the possibility of the reality of psychic phenomena.   I followed up with additional reading in the area of psychic phenomena, and discovered the scientific field of Parapsychology, and the rigorous work of researchers like J.B. Rhine at Duke University.   My thinking shifted dramatically — if psychic phenomena were genuine, and could be scientifically studied, then that pointed (at the very least) to the existence of dimensions of reality, or the operation of physical phenomena beyond our ordinary scientific and every-day experience.  Perhaps these studies would lead to a “science of the soul” .  If psychically gifted individuals can know things at a distance (without conventional sensory observation) and predict the future with results that are statistically significant), then our understanding of sensory experience, along with time and causality, needs to be revised and expanded!

This line of thinking led me to revisit my conclusions about religions and the existence of a deity.  I began reading and learning about the various religions of the world.  I had been raised in Protestant (Lutheran) Christianity, but rejected it early on when I learned about missionaries, and couldn’t see any rational basis for believing that any one religion was “more correct” than any other.  Now I became curious to learn more about religions and spirituality, with the thought that there might be some common core of truth, even if no single viewpoint was “correct in itself”.  Recall the parable of the “Blind Men and the Elephant” that I mentioned in last week’s blog post.  It seemed that a more complete understanding of an elephant could be arrived at by collecting and comparing the various blind men’s reports of “elephant experience”.  Perhaps similarly, a deeper understanding of the spiritual and supernatural could be gleaned by studying and comparing the various teachings, thinking, and experiences of various religious traditions.   I greatly enjoyed the book The World’s Religions by Huston Smith which was an excellent survey and point of departure in my explorations.

The 1960’s and 1970’s were a time of great interest, especially among the youth culture, in Eastern philosophies, traditions, meditation, etc.  The Beatles, especially George Harrison, made pilgrimages to India, and studied and practiced meditation.   I found myself similarly fascinating with “Eastern thinking”.  I especially was drawn to writings about Yoga, which seemed to offer a scientific (empirical) approach to religion.  One was encouraged to “know God” through personal experience — spiritual practice and meditation.  This was very appealing to me — I had always distrusted dogmatic pronouncements (“This is true because we said so”, or “The Bible says it, so it’s true”).   The book Autobiography of a Yogi  by Paramahansa Yogananda (published by Self-Realization Fellowship) deeply influenced me.  It is a fascinating account of Yogananda’s spiritual journey, from his early experiences to his passionate quest to find teachers (guru’s) and to grow spiritually to know God.  It led me to affiliate with SRF, and practice mediation for many years.

Another book that had enormous influence on me was Many Lives, Many Masters by Brian Weiss, M.D.  This is a book by a medical doctor (psychiatrist), steeped in the scientific world-view, who used hypnosis in his therapy practice, and stumbled upon the phenomenon of past-life regression.  His conventional scientific paradigm could not account for the phenomena he was observing, that patients could regress to past-lives through hypnotic suggestion, and that the recollection of past-life traumas could lead to nearly instantaneous remission of present-life symptoms.  Among other things, this book reinforced my growing belief that reincarnation was genuine, and that we have a soul-consciousness that survives physical death – a notion I was first exposed to in reading about Edgar Cayce.  I believe that a truly open-minded scientist must take note when other scientists report observations that run counter to the accepted scientific paradigms.  That is, in fact, the history of scientific progress – initial rejection of data that contradicts the existing paradigm, and later acceptance of these “contradictory facts”, leading to revised or novel theories that account for the “new phenomena”.   Thomas Kuhn’s book The Structure of Scientific Revolutions is a wonderful exploration of this dynamic in the history of science – another book that I recommend highly!

Books on Philosophy

I already mentioned reading Will Durant’s Story of Philosophy, which provided me an excellent overview of the history of philosophy and various flavors of philosphical thinking.  In college, as an undergraduate, I stumbled upon E.R. Emmet’s excellent book Learning to Philosophize.  This book was about how to do philosophy,  and greatly influenced how I view the enterprise of philosophical inquiry.   Memorably, it described the practice of philosophy as one of posing deep questions,  carefully defining terms, and examining alternative proposed answers to questions.   I recall that this book got me to thinking deeply about the meaning of “meaning”, which I find a very fascinating subject (and of course involves my love of self-referential ideas).

Political Books

As with many of my generation, I was profoundly shaken by the assassination of president John F. Kennedy.   I remember marveling at how quickly (within only a few hours) the media were reporting information about the “suspect”, later identified as Lee Harvey Oswald.   It set off a slight sense of doubt as to whether we were receiving the truth through the media, but I ultimately accepted the Warren Commission conclusions at the time the report came out.  It wasn’t until later, in my college years, that I began to seriously explore alternative theories of the assassination.   There turned out to be a remarkable amount of evidence that contradicted the Warren commission, and suggested that the assassination was the result of a conspiracy.   I read many books and articles on this subject (too many to list here), and have concluded that we have been lied to by our government, and that the JFK assassination was an instance of a coup in our home country!   I believe this has profound implications for understanding our country’s subsequent history, including our involvement in Viet Nam, and our more recent “wars of empire” in Iraq and Afghanistan.   For me, the deep questions here include: “Who really controls our government?”,  “Are we being systematically manipulated by our media?”.   I plan to write much more about this in future blogs.

Recently I read two very informative and insightful books about our current political-economic situation:   Corporations Are Not People  by Jeffrey Clements, and Who Stole the American Dream? by Hedrick Smith.    These books opened my eyes to the fact that the decline of the middle class and the rise of corporations is not an accident — it is the result of a sustained and systematic campaign by Corporations to increase their political and legal power and influence.  I learned that the Citizens United decision by the Supreme Court was not an isolated thing, but was instigated by Justice Lewis Powell, who, through the U.S. Chamber of Commerce, advocated for corporations to work together to increase their influence over legislation to obtain corporate-friendly laws.  This all began back in the 1970’s when Powell worked for Nixon, and before Nixon appointed Powell to the Supreme Court.   The doctrine of “corporate personhood”,  as foolish as it sounds, and as famously and ineptly referred to by Mitt Romney in his sound bite “Corporation are people, my friend”,  only developed over time.  The Citizens United decision, further opening the floodgates to unfettered corporate influence on our elections and legislative processes, was just another small step in corporations march to amass vast political power and influence.   If Fox were reporting on this they would call it the War on the Middle Class,  but they won’t, because they are another tool of  corporations, working to manipulate the uncritical populace into further supporting the “job creators” (corporations) — this would be a funny and ironic joke, if it were not so devastatingly real and harmful to those of the “99%” (middle and so-called lower classes), not to mention harming the health of our country, our economy, and our political processes.  We are facing a crisis – and I promise to devote some posts in the future to more deeply exploring these important and critical issues.

Personal Growth

I have tried to commit my life to personal growth – to becoming the best person that I can be.  My commitment is to life-long learning and self-improvement.  I have ready many books in this area, but the single most inspiring and influential book was a work of fiction:   Musashi  by Eiji Hoshikawa.    The story (saga) of Musashi Miyamoto is a Japanese epic set in Feudal Japan, and recounts the life and development of Musashi from an angy, undisciplined, rebellious, and ambitious farmer into the finest and most expert Samurai swordsman in all of Japan.   Along the way he learns that becoming the best he can be is a lot more than developing skill and proficiency in swordmanship.  It has as much to do with humility, respect and love for others, and learning in all aspects of ones life.   I had first encountered the Musashi story through the movies – a series of three movies called The Samurai Trilogy.   I loved the epic story on many levels and saw the trilogy multiple times (I now own the set of dvds), and it was gripping and inspiring to read the book as well!

to be continued …

Notes from the Library

I Love Libraries and Books

Because I love books, it is simply natural that I love libraries.  It is easy to take them for granted, yet they are amazing and wonderful institutions.  In facts, books themselves are wonderful inventions.  Libraries gather together not only books, but other media (magazines, newspapers, reference works, and more recently music and videos).  They represent a vast collection of knowledge, information, and entertainment.  The Name of the Rose by Umberto Eco is a truly wonderful book that completely changed my thinking about and appreciation of libraries!  It is set in the middle ages, when “books” were hand-copied manuscripts, and were not only rare, but valued and treasured.  The book is a murder mystery detective story set in an old monastery.   One of the greatest tragedies was for a library to burn (with incalculable loss of irreplaceable recorded thinking and knowledge!).  I wonder what wisdom and knowledge was lost when the great Library of Alexandria burned?  Fortunately today, with modern printing, not to mention electronic recording and distribution, collected knowledge is not so vulnerable.

Home “Library”

The very first “library” I remember is the collection of children’s books available in our home.  My parents purchased the Childcraft Collection,  which was a marvelous series of volumes including many fairy tales, Aesop’s fables, poems, and short stories.  I loved having this literature read to me, and later reading it for myself!  My favorite poem was “The Highwayman” by Alfred Noyes.  I loved both the tragic story and the incredible rhythm of the words.  One of my favorite stories was the “parable” of “The Blind Men and the Elephant”, which I understood much later to convey a deep truth about perception and knowledge, and the relativity of understanding.  If you are not familiar with it, essentially there are several blind men engaged in heated dispute about the nature of “the elephant” – each argues that he is the only one who truly appreciates and understands the nature of the elephant (one likens it to a wall since he felt the elephant’s side, another to a rope after touching the tail, still another to a snake because he felt the trunk, a tree trunk from feeling the leg, and so on…).  Much later, I came to view this as a parable about various religions, and the silly, pointless arguments about which is the “one true religion” and who “really” understands the nature of God.

Our “home library” also included many Dr. Seuss books.  These were read to me often enough that I soon had them virtually memorized, especially my favorite The Cat in the Hat.  This was perhaps the first book I “read to myself” as I was learning to read.  I loved the rhythm and rhyme, as well as the incredible playfulness of all the Seuss books.   In third grade, we students had on opportunity to order and purchase books through Scholastic Book Club, and I remember acquiring and reading some wonderful books like:  Snow Treasure,  Stranger than Fiction,  and many, many more!

Fifth Grade Classroom “Library”

My fifth grade teacher had an informal library of books and puzzles.  I remember reading a few Tom Swift books she had, and then starting to buy them for myself (my first personal book collection).  Tom Swift stories combined science fiction with Tom’s wonderful inventions and exciting adventures.   Her puzzle collection included Kohner Brothers’  “Hi-Q” peg jump solitaire puzzle.  I loved playing with that, and remember how difficult it was.   Later, in college, when I discovered “macro-operators”, it became easy to solve puzzles like this by thinking about them in higher-level chunks.

Hershey Public Library

I have frequented many public and school libraries over my lifetime.  The first I remember is the Hershey Public Library, which was located in the center of town in the Community Center.  It was not a large library, but I found lots of fascinating material to read both for enjoyment and learning.  My most vivid memory is discovering the “mathematics” shelves, there.  In the Summer after 7th grade, I spent a lot of time reading as many of the math books as I could.  I particularly loved the MAA series, which seemed reasonably accessible, if still challenging.  Here is where I first learned about the various classes of numbers (integers, rationals, reals, and imaginaries).  I remember struggling to accept imaginary numbers because the conventional names suggested “they weren’t real“.   Great food for my growing mathematical appetite!   I also continued to explore science fiction – I remember reading a lot of Heinlein in those days, and I think this is where I discovered Isaac Asimov’s I Robot (and other robot stories).

Hershey Junior High School Library

   I continued to explore SciFi, reading more Asimov (especially loved Foundation Trilogy), Andre Norton, and more Heinlein.  SciFi reinforced my interest and passion for technology and science, and stretched the boundaries of my imagination.  I also developed an interest in biography, reading about many famous people.  I was especially interested in inventors, and the most memorable biography was about Nicola Tesla, the inventor of generators, motors, and alternating current.   Thomas Edison tried to suppress his ideas about A.C. but they have won out in the end. Alternating current is what supplies power to our homes and appliances, though it now needs to be converted back to DC to power computers and charge phones and other devices.

Hershey High School Library

Here I continued to learn all I could from books on math and science.  My favorite magazine was Scientific American, and of course my favorite feature was Martin Gardner’s monthly Mathematical Games column.  There were also a number of collections of Martin Gardner’s essays (Mathematical Diversions), which I also enjoyed tremendously.  Martin Gardner, as I’ve noted in previous posts, had an immeasurable impact on the growth of my passions for math and puzzles.    I also recall reading the various chess books in the H.S. collection – I not only loved puzzles, but board games as well, and developed a strong interest in chess.

MIT Libraries

MIT has a large and distributed library system, spread over many different campus buildings.  The “main library” is the Hayden Humanities and Sciences library.   I spent a lot of time here, since this is where the math books were!  I could also be found at the Barker Engineering library, where more technical books, journals, and proceedings in computer science and artificial intelligence were available.  I also enjoyed the Music library, which has a great collection of classical sheet music which I took advantage of to feed my music (piano and keyboards) passion.  I was proud when my first technical publication appeared in the MIT math journal collection:  Greene,C., and Iba,G., “Cayley’s Formula for Multidimensional Trees,” Discrete Mathematics, vol. 13, no. 1 (1975), pp. 1-11.  A tiny drop added to the “sea of knowledge”.

Lexington’s Cary Memorial Public Library

We residents of Lexington, MA, are fortunate indeed to have a truly wonderful Public Library.  I am particularly fortunate in that I live (since the divorce and move in 2002) just 1 block away from the library, so I can walk there very easily.  Cary Library has an incredible collection of resources:  books, magazines, newspapers, electronic data bases, eBooks, videos, and CD’s.   Even better is the fact that Cary Library belongs to the Minuteman Regional Library Network, and offers the ability to do a single on-line search of all the libraries in the system, and then request materials, which get delivered to the local library!  A tremendous resource and service!   I found that there are numerous advantages to borrowing books instead of buying them.   Buying books has the advantage of (sometimes) getting them more quickly, and providing as much time as you care to take to read them (but this is also a disadvantage, as I will describe).  The downside is that purchased books accumulate and take up space (I’m a pack rat, and have difficulty casting off possessions, especially books).  It also turns out that many of my purchased books sit around unread for the longest time (I estimate that I’ve only read perhaps half of the books I’ve collected over the years).  Borrowing from the library has a built in “deadline” for reading it — the “final” due date (when there are no more renewals possible).  Sometimes I don’t finish a book before I have to return it, but I then simply request it again.   For popular books (especially new releases) there are no renewals because there is a waiting list, and I often have to wait weeks or months before a request finally arrives – but there are always other things to read (and lots of other things to do!) in the meantime.

Being a music lover (especially of the blues), I have purchased a rather extensive CD collection, but I simply cannot afford to buy all the music I’d like to.  The library, especially through the Minuteman Network of libraries, gives me access to a broad and rich variety of music on CD.  It lets me explore new music (new groups, new albums and songs) to see what I like and then only purchase that which I truly love.  This all gives an entirely alternate and new meaning to the title of this post:  “Notes from the Library”. 

The staff of Cary Library are dedicated, professional, and friendly.  I am profoundly grateful to have Cary Library as a convenient resource, and personally wish to thank all the staff for their hard work and service!

[… hey, looks like I finished this week’s post a little earlier for a change, and managed to keep it a little shorter!  Stay tuned for next week…]

Divorce 2002

If my first years at MIT were peak years in my life, then 2002 was an absolute low.  In June of that year, my (now-ex) wife divorced me.  It came as a shock, though in hindsight perhaps I should have seen it coming.  The first clue was in 1998 when my wife received her inheritance following her father’s death.  One of her first reactions was to tell me “Now I don’t need you any more”.   More evidence that economic issues were the basis of our relationship in my wife’s mind.   I couldn’t believe she would think like that – even if she didn’t need me economically, there was a very strong reason to persevere in the marriage (difficult as it was) – specifically our co-parenting our 3 children.   I was always committed to trying to work things out in our marriage, and we tried lots of counseling  (individual, couples, and even couples groups).   In fact on the morning of June 6, 2002  (a day that will live in infamy in my memory, and ironically D-Day, where D could stand for “divorce”), my wife and I were (so she told me) scheduled to meet a new counselor to try working again on our marriage difficulties.  It turns out there was actually no therapy appointment, rather I was set up to have “divorce papers” served on me at that time.  I was totally shocked, and terribly upset at being lied to.  The old cliche is “it takes two to make a marriage work”, and apparently my wife wasn’t committed to that.  I remember telling my boss at Gensym (back in 1998) that I might be facing a divorce, given my wife’s attitude.  Fortunately, it didn’t happen at that time.  Later I learned that she in fact had consulted lawyers, who advised her that divorcing me immediately after receiving her inheritance might lead to her having to split it with me.  So she waited, but (whether consciously or not)  made my life miserable – perhaps trying to push me into divorcing her.  That wasn’t going to happen, because my children were so important to me that I’d put up with almost anything to maintain my close relationship with them. During those intervening years she used her inheritance money to hire an architect and have extensive renovations done on our house – which was enormously stressful, and included our having to move out to an apartment for 3 months.

I became extremely depressed that day (June 6, 2002) and the weeks following.  I was walking around in a daze.  I had to scramble to line up a lawyer to represent me, and I was in fear and terror of what would happen to me.   I didn’t know where I’d be living, how I’d manage economically (I was still working part-time at MIT, but that was more for tuition savings than for income), or most importantly what would happen to my relationship with my children (2 of which were still living at home).  I quickly sought out a psychiatrist and got on medications to help me cope.   Throughout this time, my son Aaron, who was then at MIT, tried to reassure me that this would all turn out to be a blessing in disguise, and that I’d end up much happier being out of the unhappy marriage.  He was absolutely right!  In just 2 or 3 months I was changing my attitude and looking at the bright side of things — something I learned particularly from my mother!   Fortunately I didn’t have to move out immediately (my wife intended to stay in the house), and most importantly I found a terrific lawyer that I really enjoyed working with.  Unfortunately, my 50th birthday (July 4, 2002) came at a time when I was not in any mood to celebrate, so I felt I missed out on marking that half-century milestone.

By October, I had found a great (if expensive) apartment in Lexington, very near the high school  — great for my son, David, who was attending LHS at the time.  Once I arranged for the apartment, David committed to moving with me — the choice was entirely left up to him.  I was very grateful to have the opportunity to live together with him during these years.   Sadly, if understandably, my daughter Rachel stayed with her mother, and I felt a sense of loss for the reduced level of our interaction and relationship during her middle school and high school years.  Of course I saw her as much as I could, but I missed seeing her on a daily basis, reading with her, working on homework together, and simply kissing her goodnight when she went to bed (I used to sing her lullabies when she was younger).  She occasionally stayed over, but that was inconvenient for her – since all her “stuff” was at her mother’s house.

The divorce turned out much better than I had feared.  We finalized an agreement in November, 2002.  I got enough of a settlement (mostly for my half of the equity we built up in our house) that I was ok financially — turned out that by living frugally, I could live off the income from investing my assets.  Most importantly, I didn’t have to pay alimony, and my ex-wife assumed responsibility for the kid’s college expenses (something that had always been promised by her father while he was still alive, so we had never “saved for college for the kids”).


I discovered the joys of freedom!  I was free of the constant stresses of a difficult marriage.  Economically free to pursue any career directions I wished.  And free to seek out new relationships.

I started dating even in the Fall of 2002, as I was getting re-settled and finalizing the divorce.  At first it was difficult to re-accustom to the dating world, but I ended up trying on-line dating (JDate), and met quite a number of very interesting women, some of whom are still friends.  No, I was not a JDate “success story”, but I ended up in a wonderful relationship with someone I had known even before the divorce.  But that was a bit later (2006).

In Summer 2003,  I decided to have a “make-up” celebration for my 51st birthday (half-century + 1).   July 4, 2003 was a very special Independence Day!  I was much happier by this point, and invited a number of my friends and former classmates and colleagues to celebrate with me.   We had lots of food, and had an open musical “jam session” in our back yard.   I had written a number of songs during this period (song-writing and music helped me in dealing with the emotional roller-coaster of 2002).  One of the songs I wrote and shared at the birthday bash was “Losin’ the Blues”.  I was always a blues fan, but I was feeling so happy those days that I felt it was more difficult to play and write blues songs.  I plan to share that song and others on my web site, but I’m not quite there yet, so look for it in a later post on “Musical Notes”.


My part-time teaching at ESG/MIT ended in 2004.  After that, aside from the occasional consulting gigs, I devoted myself to working on my game and puzzle interests, as well as trying to get back into basic AI research.   While there are many advantages to working at home, I missed having colleagues and co-workers to interact with (and learn from!).  I started seeking out collaborators, but it was hard to find them — academics were wary of “working with someone who technically was un-employed”.

In 2008, I went to the G4G8 conference, the 8th Gathering for (Martin) Gardner.  I neglected to mention in my earlier posts how deeply I was influenced in high school by Martin Gardner’s famous Mathematical Games column published in Scientific American.  I was so glad our high school library subscribed to  Scientific American, and I, like many other budding mathematicians and puzzle lovers, voraciously read Martin Gardner’s column every month!   The G4G8 was the first “Gathering for Gardner” that I attended. It was an expensive trip for me to go to Atlanta for nearly a week, but I’m so glad I did.   It was great to see old math and puzzle friends, and to also find new ones!  These gatherings bring together people interested in the areas of Recreational Mathematics, Puzzles, and Magic (all interests of Martin Gardner’s).   Thank you, Martin, for fanning the flames of my puzzle passion, and introducing me to so many fascinating mathematical topics!  I’m sorry I haven’t been able to meet you in person and thank you directly for all your inspiration.  At G4G8  I presented a short talk on my Target Tiling video game inspired by Tetris.  Another major highlight of G4G8, for me, was meeting an editor from Sterling Publishing, who encouraged me to submit a proposal for a puzzle book based on my Round Trip puzzles.

Becoming a puzzle book author

I submitted a proposal, which was eventually accepted, so I set to work producing several hundred Round Trip puzzles, for the book.  Although I had a computer program that generated these puzzles, I still needed to hand-solve every one to rate it for difficulty.  There were also some nitty-gritty technical issues surrounding creating images for the puzzles — turned out that screen-grabs were not of sufficiently high-quality resolution (I had modified my program to automatically display and do an image capture of puzzles).  So I turned to postscript, and learned how to do rudimentary postscript programming – I was totally surprised to discover that (.ps) postscript files were simply text files with postscript commands in them and that postscript was simply another programming language.  I learned enough postscript to format .ps files of my Round Trip puzzles, and then wrote a program to automatically generate all the .ps images in a batch!   I was fortunate that my editor (or one of the publisher’s departments) was willing to do the layout of the images – that saved me a lot of work.   I did write a 20-page introduction which you can find on my web-site just under the Round Trip Puzzles book icon.   Turned out the publisher only wanted a 4-page intro, so I put the full (what I call “expanded”) intro on my web page.   After nearly 3 years, my book Round Trip Puzzles finally appeared in January 2011.   I personally think these Round Trip puzzles have the potential to be the next Sudoku and KenKen puzzle success phenomenon — and they fill a niche of logical-spatial-geometric puzzles, which I personally (being a spatial-visual thinker) relate to.  I have tried to publish a column (for free!) in various newspapers, but with very limited success thus far.  With the right promotion, I truly believe these puzzles could take off!  I should mention that Scott Kim’s “Brainteasers and Mind Benders” Page-a-Day calendar  typically includes 12 or 13 of my Round Trip puzzles, and in some years the variant One-Way-Trip puzzles.

Becoming an iPhone game developer

When my book appeared, I mentioned to my son, Aaron, that it might be cool and fun (as well as good business promotion for the book and the puzzles) to make an iphone app based on my Round Trip Puzzles.  Aaron liked that idea, and said he’d be interested in learning how to program for iphone.  I originally was hoping to learn from an iPhone app development mini-course through MIT’s IAP (January intersession), since such a course had been offered through IAP in 2010.   Sadly, it wasn’t offered in 2011, but Aaron said we could learn on our own — he is an amazing programmer (and I consider myself to be a rather accomplished programmer), and in addition he has unbounded confidence (which I lack).  So he relocated to Boston (from San Francisco) for the month of January and we started our intensive self-study in Objective C and iPhone programming.  It took us a good month just to learn how to do basic graphics display – it really shouldn’t be that hard, but it was!  We both signed up as Apple developers to get the iOS programming and development tools.  By April, we had a working rough draft of our app, and published it on the iTunes store!   The graphics and interface were all pretty rough, but the game was pretty solid and playable (and, I think, fun!), not to mention free!  We initially got a rather negative review based on the low production quality, despite our having announced it very clearly as a rough draft with the intention of getting user feedback.   There was a 2nd review that was much more encouraging – it pointed out that the app was only a “rough draft” and that the puzzles and playability were quite good!

We decided to name our app Monorail.  By April we were ready to launch our 1.0  official first release, including 50 free puzzles, and 350 more puzzles available through In-App-Purchase.  With the help of some TapJoy promotion, we had 500,000+ downloads during July, and reached the top 10 in educational puzzles (briefly #1 in a few foreign stores) in many international iTunes stores.   Actual paid sales were a small fraction (maybe 1%) of the free downloads.   Downloads (and sales) slowly tapered off as the promotion ended, but by January 2012, we passed the 1,000,000 (free) downloads milestone.  Reviews have been overwhelmingly positive, with many reporting that “Monorail is addictive”.

I truly enjoyed collaborating with Aaron.  I know myself well enough to realize that I would never (and I never say “never” lol) have been able to create Monorail on my own.  It was another important lesson in the power (and enjoyment) of collaboration!  I really loved working with Aaron on this project, and wished our collaboration could have continued, but he decided to return to his ambitions of creating new startups (he had already created one startup, called AppJet, and sold it to Google several years earlier).

I was fortunate to find some new collaborators who helped me update Monorail .  In November 2011, we released a version that included ads in the free version, and 2 additional puzzle packs for In-App-Purchase (total of 880 puzzles in all).  Also made some improvements to  the graphics and interface.  Unfortunately, the updated Monorail did not generate enough revenue to keep them involved, so there has not been an update in well over a year.

Experiencing a “Glennaissance”?

Over the last few years, I feel I’ve been entering a period of my life where I’m (once again) becoming more productive, happy, and energized with my various creative and “work” endeavors.  I quote “work” since so much of it is truly great fun.  I’m trying to follow Steve Job’s mantra of “Do what you love!”  and have been fortunate of late to be able to do that.  The things I love are:  puzzles, programming, math, and music.  I’ve been doing a good bit of each, lately.  I even started performing (keyboards and singing) again at local Open Mics this last year.  I had written a lot of songs over the years, many of them in the period following my divorce — which was a period of intense emotions (both negative and positive), and it’s been fun to share those in small supportive venues!

My search for collaborators, though on-going, has led to a number of fruitful joint-endeavors.   I loved the iphone puzzle app Relix, which I think is wonderfully challenging, so I was very excited to be able to work with the developer in producing a sequel Relix 2 (which I like even better).   I contributed a pack of really hard puzzles for Relix 2, called the Iba Insanity Pack.  I would love to hear from anyone who has solved , or even attempted(!), any of those puzzles.  They are not for beginners (fair warning!).

More recently, I’ve been enjoying working (as a level-designer) on a new iPhone puzzle app that should appear sometime this Fall.  I’m also working with one of my brother’s former students on an Android version of my Target Tiling (Tetris-inspired) game.   It would be great if some of these ventures turned out to be financially rewarding, but even if not, I’ll be very happy to share some of my puzzle ideas with a broader audience.

I still hope to get back to AI research, but to make significant progress, I know that I need collaborators to work with.  I’ll be blogging about my AI ideas in future posts, so stay tuned, and if any of the ideas interest you, please get in touch.


Made it through a first-pass overview of my life.  Seems it mostly focused on career, with other elements tossed in.  There are lots of life arcs I’d like to review and share in more detail, including my spiritual journey, more on parenting, more on my childhood / family environment, and the books that I’ve found particularly influential over my lifetime.  I also plan to elaborate on my philosophical explorations and thinking, my ideas on AI research, programming environments (programming should be much easier!),  puzzle explorations, my musical creations, and may even venture into my personal political thinking and viewpoints.  Lots to write about — I should be kept busy for a long time!

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