boards at the second level gives rise to other cases. For example, the board in which X plays the center square and then another square results in two different boards. The other two boards at the second level each generate five new boards at the third level.

I pruned many branches from the tic-tac-toe tree by appealing to a symmetry argument: the excluded boards are merely rotations or reflections of the included ones. Symmetry seems simple to humans, but a computer must be programmed or wired to recognize it. In a world of Tinkertoy engineering, symmetry operations would require elaborate structures.

Silverman was dealing with a tree, therefore, that was many times larger than the fragment shown in the illustration. But perseverance paid off, especially when Silverman employed a computer program that analyzed the game of tic-tac-toe and discovered that a great many boards could be collapsed into one by a forced move. Suppose, for example, that two squares in a row contain O's and the third is blank. The contents of the remaining two rows are irrelevant since an opponent must fill the third square with an X or lose the game.

Silverman was delighted when he tallied up the final total of relevant boards: only 48. For each of them he noted the appropriate move by the machine. The surprisingly short list of possible board positions heartened Hillis. The group converged on Hot Springs, Silverman says, "with the list of 48 patterns and only a vague idea of how to interpret them mechanically."

( Readers who have a fanatical bent­or are stranded in airline terminals­may enjoy working out the game tree on a few sheets of paper. How long does it take, after all, to draw 48 tic-tac-toe patterns? Four symbols should help sort things out X O, blank and a dash for "don't care.")

Once settled in Hot Springs, the team assembled the raw material for

their spool-and-stick odyssey: 30 boxes of Tinkertoys, each containing 250 pieces. Some team members put together the supporting framework that would hold all 48 memory spindles. To explain precisely how the spindles were made, I must digress for a moment and describe the conventions employed by the team to encode tic-tac-toe positions.

First, the squares of a tic-tac-toe board were numbered as follows:

1 2 3
4 5 6
7 8 9

Then a memory spindle was divided conceptually into nine consecutive lengths in which information about the status of each tic-tac-toe square was stored from left to right.

Each length was further subdivided into three equal sections, one for each possible item one might find in a square: an X, an O or a blank. Each possibility was encoded by the lack of a spool. For example, if an X happened to occupy a certain square, the memory spindle would have no spool in the first position, one spool in the second and one spool in the third. Similarly, a spool missing in the second position denoted an unplayed square, and one missing in the third position symbolized an O. Finally, if all three spools were missing, it meant that what occupied the square was irrelevant.

One can hardly mention the subject of memory spindles without bringing up the core piece, a thing of digital beauty. Here the Latin digitus came into its own, the construction resembling a kind of rotating claw with nine fingers. The core piece and a sample memory spindle are shown in the illustration below.

The core piece consisted of nine equal sections. Each had its own finger, a short stick protruding from the rim of a sliding spool. Within each section the finger couid be moved

along the axis of the core piece into any of three possible positions: one for X, one for O and one for blank. The core piece could therefore store any possible tic-tac-toe board by virtue of the positions of its nine fingers as moved by the operator for each play by human or machine. In the illustration below, fingers in the consecutive positions 2,1, 2, 3,1, 2, 2, 2, 2 would represent the board shown.

If the current situation of play is stored in the core piece, does the Tinkertoy computer require any other memory? Could spool-and-stick logic devices be strung together to cogitate on the position and ultimately to signal a move? Well, yes­but such a Tinkertoy computer would be complicated and immense. The memory spindles eliminated the need for most of the computer's cogitation. All the Tinkertoy computer had to do was to look up the current board in the memory spindles. The only purpose of the search, naturally, was to decide what move to make.

A glance at the illustration on the preceding page makes it clear that each memory spindle was accompanied by a number written on a paper strip hanging next to its output duck. These numbers were the machine's responses. As the read head clicks down the rows of spindles, the core piece wants to turn but cannot as long as at least one memory-spindle spool blocks one of the core piece's nine fingers. Only when the read head falls adjacent to the spindle that matches the current board do all nine fingers miss. Then the core piece whirls.

By a mechanism that would do Rube Goldberg proud, a stick protruding from the end of the core piece engages another stick connected to the output duck. The spinning core piece thus kicks the duck off its perch to peck at a number writ large on the paper strip.

Computer purists will ask whether the Tinkertoy contraption really deserves the title "computer." It is not, to