Pairwise versus overall

A recurring general theme in mathematics is of a property being true of three objects when considered pairwise, but not true of the entire set. For example, earlier we considered non-trivial mutual friends, and how a simple graph is not sufficient to contain all information about the relationships between elements of a set. Here are a few more examples:

Coprimality

The integers 6, 10 and 15 are coprime, but any pair of them share a common factor. Consequently, the Diophantine equation a^6 + b^{10} = c^{15} cannot be solved by the Sam Cappleman-Lynes technique, nor can it be disproved by Fermat’s Last Theorem.

In the statements of Beal’s conjecture, Fermat-Catalan conjecture and the abc conjecture (proof still unverified at the time of writing, since precisely one person understands inter-universal Teichmüller theory.) it doesn’t matter whether we use coprimality or pairwise coprimality, since the relation between the variables causes the notions to coincide.

Probabilities

If an ordered triple of random variables (x, y, z) can take the values (0, 0, 0), (1, 1, 0), (1, 0, 1) and (0, 1, 1) with equal probability, then any pair of variables are independent, but the third is determined entirely by those two. A similar situation occurs in three-particle quantum entanglement, where the total parity of the electron spins is restricted by conservation laws.

Food incompatibility triad

George Hart (geometer, and co-creator of Vi Hart) proposed the following problem:

Find three foods, such that any two are compatible, but the three together are incompatible.

This has been fiercely debated in the comments section of an article in the Guardian column of Alex Bellos. There doesn’t seem to be any generally accepted solution (Feynman erroneously claimed that {milk, tea, lemon} was a solution, but this is broken by the incompatibility of milk and lemon), and little progress has occurred since George Hart closed the question on the basis of it being too subjective.

Until now, that is. John Howe, Cameron Ford et al claim the following solution:

  1. Pancakes
  2. Lemon liqueur
  3. Beef

The first two constitute a crêpe suzette, and the other two combinations apparently exist on the Internet, whereas their union does not. The solution has apparently been sent to George Hart, and I am unaware whether a response has been received. This may finally put to rest an age-old puzzle believed to have originated with the philosopher Wilfrid Sellars.

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