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I can't help thinking that the solutions thus far proposed are ignoring this ....
" Every evening a evening a gaoler would visit them in turn, and if either of them could tell the gaoler the total number of towns in the kingdom, they would both be released that same evening. But if either of them gave the wrong answer, they would both be executed. "
How can they proceed to the next possible solution without knowing their previous one was wrong ? Presumably, also, they cannot communicate with each other through locked doors ?
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