Puzzling End for the World?
A French mathematician whose name crops up frequently in books on recreational mathematics is Edouard Lucas. His best known invention, the Tower of Hanoi, appeared in 1883 under the name of M. Claus of the College of Li-Sou-Stian (anagrams of Lucas and the Lycée Saint-Louis where he taught).
A year later De Parville provided an alternative name for the puzzle, the Tower of Bramah, and an imaginative legend to explain its origins.
In the great temple at Benares there is a brass plate in which are fixed three diamond needles. At the creation God placed sixty-four discs of pure gold on one of these needles, arranged in order of size, the largest disc resting on the brass plate. Day and night the priests move the discs, one at a time, from one diamond needle to another according to the fixed and immutable laws of Bramah, never placing a larger disk on a smaller one. When all sixty-four discs have been transferred from the needle on which God first placed them to one of the other needles, then tower, temple, and Brahmins alike will crumble into dust, and with a thunderclap the world will vanish.
It is quite easy to make a simple version of the Tower of Hanoi using coloured paper or cardboard. Cut out four rectangles and a base for the puzzle, then lay them out on a table or desk as shown below.
The challenge now is to move the pieces one at a time, never placing a larger piece above a smaller one, until you have moved the whole ‘tower’ to a new position.
If you can solve this version of the puzzle, you are ready to answer some questions.
- What is the least number of moves required to transfer the tower to a new position?
- If you want to transfer the tower from A to C, what should your first move be?
Can we use our experience with a 4-disc tower to predict when the world will end? Perhaps we can, but it might help to try the puzzle with other numbers of discs. If we make a table of results we might see a pattern.
Number of discs, n | 1 | 2 | 3 | 4 | 5 | 6 |
Least no. of moves, Tn | 1 | 3 |
By looking at the way you do the puzzle you might be able to write down a formula connecting Tn and Tn+1. For instance, when solving the 6-disc puzzle you might be able to see the connection between T5 and T6.
(The picture below, showing how the puzzle might look part way through the solution may give you a clue. )
By now you should have found a rule for using one answer in the table to predict the next, but this method for working out the least number of moves for the 64-disc tower will take a long time.
Look again at the numbers T1=1, T2=3, T3=.., T4=..
There is a general formula for Tn in terms of n. Can you find it and explain why it works?
Using this formula, it turns out that the number of moves needed to transfer the 64 discs of the Tower of Bramah is :
18 446 744 073 709 551 615
If the Brahmins move one disc per second, how many centuries will it be before the world ends?
