Twenty-six villages are ranged around the coastline of an island. Their names, in order, are A, B, C, …, Z. At various times in its history, the island has been visited by 26 missionaries, who names are also A, B, C, …, Z. Each missionary landed first at the village that bore his name and began his work there. Each village was pagan to begin with but became converted when visited by a missionary. Whenever a missionary converted a village he would move along the coastline to the next village in the cycle A–B–C-…-Z–A. If a missionary arrived at an uncoverted village he’d convert it and continue along the cycle, but there was never more than one missionary in a village at a time. If a missionary arrived at a village that had already been converted, the villagers, feeling oppressed, would kill him and revert to a state of paganism; they would do this even to a missionary who had converted them himself and then traveled all the way around the island. There’s no restriction as to how many missionaries can be on the island at any given time. After all 26 missionaries have come and gone, how many villages remain converted?
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None. Each village is converted by its namesake missionary, unless it has already been converted by someone else, so every village gets converted at some point. But then each becomes a deathtrap for the next missionary who visits it. As each missionary works his way around the island, he leaves a string of such traps behind him, the first of which will spring on him when he comes around again unless he’s killed even sooner by stumbling on a village converted by someone else. So all 26 missionaries must die. Since a murderous village reverts to a pagan state, over time the 26 deaths provide 26 unconverted villages, but these are not necessarily different. If we could show that once a village has killed a missionary it is never visited again, and is thus effectively removed from the story, then the 26 deaths leave behind 26 different unconverted villages.
So suppose village A has just murdered its first victim. That means it has been visited twice, once by the missionary who converted it from paganism and then by the missionary whom it murdered. What is the possibility that it will receive a third visit? Well, one of the two known visitors might be missionary A on his first visit to the island. But any other visitor to A must come from village Z. So if A were to get 3 visitors, then at least 2 of them must have come from Z; that is, at least 2 missionaries must have got into and out of village Z. But Z kills every other missionary who visits it. So if 2 missionaries left Z, then at least 3 must have entered it. Thus in order for A to receive a third visitor, Z must already have had at least 3 visitors. By the same reasoning, Y must have had at least 3 visitors, and so on around the island, until we reach the impossible condition that in order to have a third visitor, village A must already have had 3 visitors. This shows that A cannot receive a third visitor, and the conclusion follows.
(From Crux Mathematicorum, via Ross Honsberger’s More Mathematical Morsels, 1991.)
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