[This is post #3 in a series on the Thue-Morse sequence, which was introduced two weeks ago and discussed further last week.]

While the Thue-Morse sequence appears in many branches of Mathematics^[1], this sequence has also captured a place in the history of chess.

With only the basic rules of chess (specifically, how pieces move and capture and how a game may end by checkmate or stalemate), there is no guarantee that a game of chess will ever finish. For example, two (very bored!) players could theoretically move their knights back and forth ad infinitum without ever engaging each other or capturing a piece (but why would they want to?). A more realistic example is drawn below. In this game, black will soon checkmate white if given the freedom to move, and white has no way to win. However, white can avoid losing by keeping black in an unending runaround with repeated checks, as illustrated. In this case (a phenomenon known as perpetual check), forcing an infinite game is more advantageous for white than allowing an otherwise inevitable loss.

Because of this possibility of never finishing, a search began for new chess rules to guarantee that every game of chess eventually ends. One suggested rule was:

A chess game ends with a draw if a sequence of moves—with all pieces in exactly the same positions—is played three times successively.^[2]

The hope was that any sufficiently long chess game would eventually repeat itself enough to break this rule. However, as Mathematician and Chess Grandmaster Max Euwe showed in 1929^[3], this hope is not correct. Indeed, there exist infinite chess games that never repeat the same block of moves three times in a row!

How did he prove this? With the Thue-Morse Sequence, of course! He proved the very unintuitive fact that the Thue-Morse sequence has no chunk repeated three times consecutively. Such a thrice-repeated sequence is called a cube, so this property can be restated by saying that the Thue-Morse sequence is cube-free. For example, you will never find the sequences 1 1 1 or 011 011 011 in the Thue-Morse sequence. (Challenge: show that these two sequences do not appear. Harder challenge: show that the Thue-Morse sequence is cube free!) So, if the “very bored” knight-flippers described above decide to move their right knights or left knights according to the 1s and 0s of the Thue-Morse sequence, the resulting game will have no tripled move sequences.

Luckily, the above rule is not part of the current official chess rule set. There are now two rules on the books that can be used to eventually end any chess game. (In fact, either one would suffice. [Prove this!]) A draw may be declared in either of these situations:

• The exact same board position appears three times ever during a game. (Every detail must be the same, such as whose turn it is, which kings have the ability to castle, etc.)
• No pawn is moved and no piece is captured in 50 consecutive turns. (A turn consists of one move by each player.)

However, there’s (always) a catch. These two rules only take effect if a player notices and decides to declare a draw. So even with all modern rules in effect, the bored knight-flippers are still allowed to play their infinite game of chess—cube-free or not.

Notes

1. A great survey called The ubiquitous Prouhet-Thue-Morse sequence by Allouche and Shallit documents many sightings of this sequence throughout Mathematics, including Combinatorics, Differential Geometry, Number Theory, Analysis, and as discussed here, Combinatorial Game Theory. [↩]
2. Manfred Börgens, Max Euwe’s sequence [↩]
3. in his paper titled Mengentheoretische Betrachtungen über das Schachspiel [↩]