Solution: The unfair coin
The example is due to John von Neumann, taken from his 1951 paper Various Techniques Used in Connection with Random Digits
The experiment consists of two tosses of the biased coin. If the result is HT
you agree to count it as “heads”; if the result is TH you agree to count it as
“tails”, or vice versa. If the result is HH or TT you ignore the result and
start over. This is equivalent to a fair toss under the assumption that the
outcomes of the successive tosses are independent.