In Circularity, Ron Aharoni mentions a story by Raymond Smullyan. On a certain island there are two kinds of people, those who always lie and those who always tell the truth. One day an islander is arrested on suspicion of murder. At his trial he says, “The murderer is a liar.”

Smullyan argues that this piece of evidence alone should acquit him. If the man is honest, then what he says is true, the murderer is a liar, and since he himself is a truth-teller he cannot be the guilty party. On the other hand, if he’s a liar then his testimony is false, which means that the murderer is in fact not a liar, and once again he cannot be guilty. Either way, he proves his innocence by showing that the murderer and himself belong to two different tribes.

Aharoni adds, “The problem is that the man was found beside the corpse with a bloody knife in his hand and a wide smile on his face. He is obviously the murderer, which means that he managed to prove an obvious fallacy. It seems that using his method, he can prove anything. And indeed he can. See what he is claiming when stating that the murderer is a liar: ‘If I am the murderer, then I am a liar’, which means ‘if I am the murderer then this is a lie’. In other words — ‘If I am the murderer then L is true’. And … this proves that ‘I am not the murderer.'”

Write down a 0:

0

Now mentally “flip” this string in binary, exchanging 0s and 1s, and append this new string to the existing one:

0 1

Keep this up and you’ll get a growing string of 0s and 1s:

0 1 1 0
0 1 1 0 1 0 0 1
0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
…

This is the Thue–Morse sequence, named after two of its discoverers, Axel Thue and Marston Morse. One interesting property of the sequence is that, no matter how far it’s extended, it contains no “cubes,” no instances in which some nonempty string occurs three times in a row. For example, the last line above contains both 11 and 00 but no instance of 111 or 000. (It also contains 1001 twice in a row, but not three times.)

Max Euwe, the Dutch mathematician who was world chess champion from 1935 to 1937, used this principle to show that chess was not a finite game. Under the rules at the time, a chess game would end in a draw if a sequence of moves (with all pieces in the same positions) were played three times in a row. Euwe used the Thue-Morse sequence to show that this need never happen: If 0 represents one set of moves, and 1 represents another, and each set leaves the board position unchanged, then the Thue-Morse sequence shows that two players might step through these routines forever without ever playing one three times in a row.

Modern chess rules have dropped the threefold sequence provision. Instead a draw results when the same board position occurs three times, or when 50 successive moves occur without a capture or a pawn move. Both of these rules limit a game to a finite length (although one player must actually claim the draw).

(Thanks, Pål.)

In 1986 British electronics engineer Lee Sallows invented the alphamagic square:

As in an ordinary magic square, each row, column, and long diagonal produces the same sum. But when the number in each cell is replaced by the length of its English name (25 -> TWENTY-FIVE -> 10), a second magic square is produced:

Now British computer scientist Chris Patuzzo, who found the percentage-reckoned pangram that we covered here in November 2015, has created a double alphamagic square:

Each row, column, and long diagonal here totals 303370120164. If the number in each cell is replaced by the letter count of its English name (using “and” after “hundred,” e.g. ONE HUNDRED AND FORTY-EIGHT BILLION SEVEN HUNDRED AND TWENTY-EIGHT MILLION THREE HUNDRED AND SEVENTY-EIGHT THOUSAND THREE HUNDRED AND SEVENTY-EIGHT), then we get a new magic square, with a common sum of 345:

And this is itself an alphamagic square! Replace each number with the length of its name and you get a third magic square, this one with a sum of 60:

Chris has found 50 distinct doubly alphamagic squares, listed here. I suppose there must be some limit to this — is a triple alphamagic square even possible?

(Thanks, Chris and Lee.)

Many thanks to reader Colin White for this:

In her 2005 song “π,” Kate Bush sings the number π to its 78th decimal place, then jumps abruptly to the 101st and finishes at the 137th.

The BBC’s More or Less advanced the “Kate Bush conjecture”: that the digits that Bush sings are contained somewhere in the decimal expansion of π — just not at the start.

The conjecture is true if π turns out to be a “normal” number, meaning essentially that all possible sequences of digits (of a given length) appear equally often in its expansion.

π hasn’t been proven to have this property, though it’s expected to be the case. So, for now, “The Kate Bush conjecture is plausible but unproven.”