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Today Puzzle #621

Puzzle No. 621 鈥� Wednesday 28 November 2019

Beaver, the mascot of St Anne’s College, Oxford, knows that St Anne’s, as one of the first colleges to support women’s education, has a rich tradition of extending educational opportunities. He dreams of securing a place at the College’s top rated nursery and is keen to prove his worth against the other tiny tots, whose parents want them to benefit from the nursery’s forest school activities. When Beaver turns up for interview, he is taken to a large room full of children – 24 in total! Beaver is told they will play a game, whereby all the children, and Beaver himself, will sit in a circle, in seats numbered in order from 1 to 25. It is an elimination game that works as follows: starting from seat 1 and going around in order, every second person among those who remain in the game is eliminated. The winner is the last remaining individual.

Beaver knows to avoid seat 2 because that will be the first one eliminated. Where should Beaver place himself to ensure he wins the game, thus earning him the respect of his peers and guaranteeing his spot in the nursery?

Today’s #PuzzleForToday has been set by Junaid Mubeen Director of Education at Whizz Education and St Anne's College alumnus.

Click here for the answer

If you count along two seats at a time, you'll find that the last remaining seat is 19.

For a more general argument, you may notice that when there are 2^n seats, the person in the first seat wins (you can formally prove this by induction). So Beaver needs to be in the first seat at the point when there are 16 people remaining (16 is the first power of 2 less than 25). That means 9 people have to have been eliminated, which takes up the first 18 seats, leaving Beaver to take his place in the 19thseat.

This puzzle is a variant of the Josephus problem and can be explored with different numbers of seats and skips (e.g. every third person eliminated until two remain).

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