Solution
Eight Claims About the Tally
Show the problem again
Eight statements are written on a card, and each speaks about $T$, the total number of true statements on the card (including itself). 1. $T$ is a prime number. 2. $T$ is a perfect square (0 counts as one). 3. $T$ is even (0 counts as even). 4. $T$ is at least 5. 5. $T$ is a multiple of 3 (0 counts as a multiple of 3). 6. $T$ is at most 3. 7. $T$ is 6 or 8. 8. $T$ is a multiple of 4 (0 counts as a multiple of 4). Each statement is true or false according to its content. What is $T$?
Worked solution
Whatever $T$ is, it lies between 0 and 8, and the truth of every statement depends only on $T$. So test each candidate: count how many statements it would make true, and keep $T$ only if that count equals $T$ itself. The counts are: $T=0$ gives 5 true statements, $T=1$ gives 2, $T=2$ gives 3, $T=3$ gives 3, $T=4$ gives 3, $T=5$ gives 2, $T=6$ gives 4, $T=7$ gives 2, $T=8$ gives 4. Only $T = 3$ is self-consistent: with $T = 3$ the true statements are exactly 1 (3 is prime), 5 (3 is a multiple of 3) and 6 (3 is at most 3), which is three statements. The answer is 3.
Source: Original problem written for AxiomIQ, in the tradition of self-referential truth-counting puzzles.