Suppose a (fair) coin has been tossed 20 times. The first 10 tosses came up heads, then seven tosses came up tails, and three tosses came up heads. What is the probability that the next toss is heads?

You might think that there should be little or no chance of heads, because a few tails are now ‘due’ or even ‘overdue’ until we get roughly equal numbers of heads and tails.

But is the coin supposed to ‘remember’ how many times it came up heads or tails? What if someone had seen only the last 10 tosses (of which seven were tails); that person would be expecting a head, while someone who saw only the last three tosses (which were all heads) would be expecting a tail. Should the coin ‘know’ how long it had been observed, and somehow change its behaviour to match what the observers are expecting? What if different observers have different expectations (or are hoping for different outcomes, if they made bets on the coin’s behaviour)? It is much simpler for the coin to have 50 per cent chance of coming up heads on each toss, and 50 per cent tails, no matter what happened in the last three (or 10 or 20) tosses.

Mathematicians also have expectations that sometimes turn out to be false, and not just about coins. In 1919, George Pólya conjectured that for any natural number n, at least half of the natural numbers 1, 2, ..., n have an odd number of prime factors. This was found to be true for all numbers up to a million. It is even true for numbers up to 900 million (these are large numbers, but they only have nine digits). However, in 1958, Haselgrove proved that there is some number (with around 361 digits) for which Pólya’s conjecture fails.

An explicit counterexample for the number 906,180,359 was given by R. Sherman Lehman in 1960; in 1980, Minoru Tanaka found a counterexample for the slightly smaller number 906,150,257, and we now know that that is the smallest counterexample.

We also know that there are infinitely many natural numbers for which Pólya’s conjecture is false, and infinitely many for which it is true.