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  1. +217 −0 counting-t4-configurations.markdown
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+title: Counting t4 configurations
+author: Carl Mäsak
+created: 2012-04-25T15:37:54+02:00
+<img style="display: block; margin-left: auto; margin-right: auto; padding: 1em" title="Photo by prilfish," src="" />
+I discovered the t4 task in a beach paradise on the island Ko Lanta in
+Thailand. I was there with friends, having a long-awaited December vacation,
+away from dreary Swedish cold and dark and snow. Instead, we were enjoying
+sun and shade, cool drinks on the beach or the quiet of our air-conditioned
+condo, getting caught up with reading, talking amongst ourselves, and exploring
+our surroundings. We seemed to have gotten there just a few weeks before the
+normal wave of tourism, so we felt uniquely at peace and at ease.
+One day I was playing around with my Android phone, looking for some game to
+distract me in this oddly tranquil place. Since I'm ever fascinated with games
+on hexagonal grids, I searched for "hex". And I found this hex-grid-based
+one-player puzzle game that had me instantly hooked.
+The easiest way to understand the t4 task is probably to [play the
+and actually drag some pieces around.
+<img style="display: block; margin-left: auto; margin-right: auto; padding: 1em" title="Hex Slide 1,000 by Tiny Bite Games, LLC" src="" />
+The description of t4 merely captures in formal writing what the puzzle game
+shows plainly:
+ This problem makes use of a board of hexagonal cells that looks like this:
+ i1 i2 i3 i4 i5
+ j1 j2 j3 j4 j5 j6
+ k1 k2 k3 k4 k5
+ l1 l2 l3 l4 l5 l6
+ m1 m2 m3 m4 m5
+ n1 n2 n3 n4 n5 n6
+ o1 o2 o3 o4 o5
+ Each cell has up to six neighbors. l3 has neighbors (clockwise) k3, l4, m3,
+ m2, l2, and k2.
+ The board is populated with some number of straight pieces of length 2 and 3.
+ Pieces never overlap, but they can move in the direction of their length,
+ which we will refer to as a "groove". So for example, a piece that starts out
+ on locations l1 and l2 can "slide" over to rest on locations l5 and l6.
+ No valid move can make a piece leave the groove in which it was first found.
+ We write "groove" because "row" doesn't quite capture how the pieces can be
+ situated. Besides the above seven rows of the table, a groove (and thus the
+ pieces in it) can also run along a diagonal direction, like this:
+ e1 d1 c1 b1 a1
+ f1 e2 d2 c2 b2 a2
+ f2 e3 d3 c3 b3
+ g1 f3 e4 d4 c4 b4
+ g2 f4 e5 d5 c5
+ h1 g3 f5 e6 d6 c6
+ h2 g4 f6 e7 d7
+ Or a groove and its pieces could run along the other diagonal direction:
+ p2 q4 r6 s7 t7
+ p1 q3 r5 s6 t6 u6
+ q2 r4 s5 t5 u5
+ q1 r3 s4 t4 u4 v4
+ r2 s3 t3 u3 v3
+ r1 s2 t2 u2 v2 w2
+ s1 t1 u1 v1 w1
+ There are a total of 23 grooves on the board, 7 horizontal, and 8 for
+ each diagonal. Grooves vary in length from 2 (out in the corners) to
+ 7 (around the main diagonals).
+The t4 description goes on for some time, but I'll stop there for now, because
+this post doesn't consider the dynamic aspects of the puzzle, only the board
+Parenthetically, figuring out a decent coordinate system for the board, which
+manages to encode the fact that the grooves constrain the pieces, and the way
+each location on the board is the intersection of (exactly) three grooves, took
+me somewhere between hours and days of thinking. I was a bit disappointed to
+have to make that part public (in order to rein in the solutions enough and to
+be able to write base tests for the task). Looking at the solutions sent in,
+I don't think many people appreciated how much one can actually get for free
+with this particular encoding.
+The t4 task is to write an algorithm that gets one special piece from one side
+of the board to another. Sometimes other pieces will block its way, and will
+have to be moved away first. And so on, recursively. The "and so on" bit is
+what makes this an interesting problem.
+My attention quickly focused on more basic matters, though.
+The app on my Android phone sports a thousand distinct puzzles. Presumably the
+good people at [Tiny Bite Games]( generated these
+algorithmically somehow. Some of these are on slightly different boards, but
+I've chosen to ignore those for the purposes of t4. There's also a non-free
+version with ten thousand puzzles.
+How many puzzles are there?
+Or, to be precise, how many board configurations are there? By "board
+configuration", I mean every legal way to strew pieces on the board, everything
+from leaving the board empty to completely filling it up with length-2 and
+length-3 pieces. All of them.
+<img style="display: block; margin-left: auto; margin-right: auto; padding: 1em" src="" />
+As we headed out from Ko Lanta, I tried to upper-bound this, with just pen and
+paper, just multiplying out all the ways to place length-2 and length-3 pieces
+in all the grooves but without accounting for collisions. I arrived at a
+ridiculous number of combinations: about 3e57. I looked out the window of the
+taxi-jeep and saw my first real-life elephant. It looked small in comparison.
+Now, "all possible board configurations" isn't the only number that might
+interest us here. It's just the "universe" of objects that we're dealing with
+in t4. As days and weeks went by (and I eventually got back home from
+vacation), finding the exact number of configurations became a worthy
+challenge, a Mount Everest of sorts.
+But there are other, smaller numbers, which are also interesting:
+1. The number of configurations with a piece l12. ("start-configurations")
+2. The number of configurations with a piece l12 *and no other l-groove piece*.
+3. The number of solvable start-configurations.
+4. The number of trivially solvable start-configurations (where the solution is
+simply `l[12 -> 56]`).
+But I didn't want to attack these problems until I had felled the beast.
+Finding the number of all possible configurations, the size of the problem
+space, the extent of the universe. Which was hopefully a lot smaller than 3e57.
+Here's one way to solve it, in pseudocode:
+ Set confs <- 0
+ For all combinations of piece placements
+ If there are no forbidden overlaps
+ Set confs <- confs + 1
+ Output confs
+It's a wonderfully simple program to think up. And it only has one loop in it.
+Let's say we happen to have an unbelievably fast computer at hand, which runs
+the loop in an optimized way so that each iteration takes a nanosecond on
+average. That's faster than today's computers, for sure.
+Yeah, so. You run that on your fast computer, and I'll go brew some coffee.
+See you in nine and a half million million million million million million
+So, that clearly wouldn't work. The program was simple to implement, but it
+just wouldn't terminate before our physical universe did. Basically, if you
+have to loop over ~3e57 things, you're screwed.
+I needed something that could intelligently weed out impossible branches as it
+went along, that didn't make the stack or the RAM blow up. Come to think of it,
+something a little like the [Dancing
+Links]( algorithm, which computes
+solutions to "[exact cover]("-type
+problems quickly (considering), and in constant space.
+Conveniently, I had written a DLX solver in 2011. In C, no less (for speed).
+The idea with that project, which was only partly realized, is to make it very
+easy to specify problems in some "human" representation, and then frontends and
+backends to the solver would take care of the translation.
+Of course, the board configuration enumeration problem wasn't an exact cover
+problem, I knew that. It would be if I was only interested in all the possible
+ways to fill up the board completely with length-2 and length-3 pieces. (There
+are 11,071,306 such configurations. Because I knew you were wondering.) But I
+didn't want that, I wanted board configurations with "gaps" in them too.
+(Especially since these are the only ones that can actually be solved!) But I
+figured I could extract the "essence" of the DLX algorithm, which is to be
+clever about possible alternatives at each choice point.
+<img style="display: block; margin-left: auto; margin-right: auto; padding: 1em" src="" />
+After a week of trying and failing to write an algorithm that borrowed the
+"essence" of the DLX algorithm, I came to an embarrassing realization.
+The problem I wanted to solve *is* an exact cover problem. It can be solved
+directly using the DLX code I already had.
+It was the word "gaps" that had me confused. Exact cover problems are
+characterized by the fact that their solutions have no overlaps, and no gaps.
+(That's what the "exact" in "exact cover" means.) But, fine, let's replace the
+word "gap" by the word "marshmallow". Now, what I was looking for was all the
+possible ways to fill the board with length-2 pieces, length-3 pieces, and
+marshmallows. Voilà, exact cover problem.
+Some problems are made solvable simply by restating them. Also, don't
+underestimate the power of reifying nothingness.
+So, I fed a representation of the problem into my DLX solver. It ran for two
+weeks on a decent computer, enumerating millions of configurations a second. It
+came up with this answer:
+There are 4,783,154,184,978 board configurations. A bit short of five million
+million. Graah, I felled the beast!
+Before this long calculation was even finished, it had occurred to me that (1),
+(2) and (4) on my list were *also* describable as exact cover problems. Only
+(3) is tricky and requires actually making moves on the board. But the other
+three questions can be formulated as exact cover problems by changing the shape
+of the board, essentially "forbidding" either one groove to occupy a location,
+or forbidding the location outright.
+I started generating such exact cover models with a [100-line Perl 6
+script]( The numbers that fell out were these:
+1. There are 573,538,221,334 configurations with a piece l12.
+2. There are 375,873,151,406 configurations with a piece l12 and no other l-groove pieces.
+3. *The number of solvable start configurations is still unknown as of this writing.*
+4. There are 7,767,954,496 trivially solvable configurations.
+At least now it doesn't feel unlikely at all that some game company manages to
+generate ten thousand hex puzzles.

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