Editing Openai/694fa129-5c4c-800e-b887-30cd25632750 (section)

=== Assistant: Write the three primes as digit-strings A,B,CA,B,CA,B,C and look at the string ABCABCABC. ===
Write the three primes as digit-strings A,B,CA,B,CA,B,C and look at the string ABCABCABC. If that string is a palindrome, reading it backwards gives

reverse(C) reverse(B) reverse(A)=A B C.\text{reverse}(C)\,\text{reverse}(B)\,\text{reverse}(A)=A\,B\,C.reverse(C)reverse(B)reverse(A)=ABC.
For that equality to hold the middle block must read the same forwards and backwards, so BBB itself is a palindromic prime. (Otherwise the reversed middle block would put different digits into the centre of the whole string.)

So BBB must be a palindromic prime (the only two-digit palindromic prime is 11; single-digit primes are palindromes too). The outer blocks must be digit-reverses of one another (their digits mirror around the centre). That means you cannot have one outer prime be single-digit and the other two-digit (a two-digit number’s reverse is two-digit), so the obvious choice for distinct outer primes is a pair like 13 and 31.

Jane’s proposal 3,11,133,11,133,11,13 violates that forced digit-structure: a single digit 3 cannot be the digit-reverse of the two-digit 13, so it cannot be the correct triple arising from the palindrome condition Bob stated. Hence Bob is justified in rejecting her guess.