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Tower of Brahma


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A horizontal board has three vertical pegs. Five discs, of different sizes and each with a hole in the middle, have been placed on one of the pegs with the largest disc on the bottom and the smallest on the top. The problem is to transfer all the disks to one of the other pegs moving only one disc at a time and never placing a larger disc on a smaller one.


ABC

Solution





Additional information

You can introduce the exercise by this re-telling of a legend (adapted from deParville, La Nature, Paris 1884) in which the 'Tower of Brahma' is used as a countdown to the end of the world!

In the great temple of Benares, beneath the dome which marks the centre of the world, rests a brass plate in which there are fixed three diamond needles, each a cubit high and as thick as the body of a bee. On one of the needles, at the time of creation, God placed sixty-four discs of pure gold, the largest disc resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the Tower of Brahma. Day and night unceasingly the priests transfer the discs from one diamond needle to another according to the fixed and immutable laws of Brahma, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc beneath it. When the sixty-four discs have been thus transferred from the needle on which at creation God placed them to one of the other needles, tower, temple and Brahmins alike will crumble into dust, and with a thunder-clap the world will vanish.

If you are worried, you will be pleased to know that even if the priests could move the discs at the rate of one a second without making a mistake, it would take nearly six billion centuries to complete the task! Nostradamus was not so optimistic.

The minimum number of moves required to move the discs is 2n-1 where n is the number of discs. In our puzzle there are five discs so the task will take a minimum of 25-1 or 31 moves.


Variations

This task can be undertaken using small discs or playing cards (Ace–5 to represent the discs and three other cards placed face down to represent the pins) or you can make giant sets of discs and pegs to be used as an outdoor group exercise.

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