Three needles are fixed on a flat plate. On one of these needles are placed sixty-four discs, the largest disc resting on the plate and the others getting smaller and smaller up to the top one. The objective is to move all the discs from one needle to another subject to the following rules. You must only move one disk at a time, and there must never be a smaller disc below a larger one during the transfer. What is the minimum number of moves required to complete the transfer of all the sixty-four discs? (Hint: Start with two discs and then three and so on. Do you see a pattern?)
This puzzle is popularly known as the Tower of Brahma (Tower of Hanoi). Instead of providing a solution, it is interesting to look at another related puzzle which has a lot of similar elements. In fact, it is not much of a puzzle as it is a hint to the previous question.
There was a king who was so pleased with his minister for teaching him the game of chess that he offered him a reward of his choice. The minister asked the king to give him one grain of wheat to put on the first square of the chessboard, two on the second square, four on the third square and so on doubling the number of grains for each square all the way up to the 64th square. The king agreed to his request while silently enjoying the thought that his minister has turned down a golden opportunity to ask for a part of his kingdom or treasury. But he soon realized his folly and had to behead the minister to avoid the shame of being indebted to him.
Well, it turns out that the number of grains of wheat the king had to give the minister equals the number of moves required to transfer the discs from one needle to another and it is such an astronomically huge number1 that the king was left with no other choice. It is said that Lord Brahma placed sixty-four golden discs on diamond needles in a temple in Benares at the moment of Creation. And when all the discs are transferred to another needle, the world along with all his creations will come to an end.
While these two puzzles are relatively simpler, there is another problem which a shopkeeper near my house once asked me. Though I figured out the solution after much effort, I still don’t have any kind of a proof which would take me to the answer. Any help in this regard would be much appreciated. Here’s the question:
He had a weight of mass 40kg which fell down one day and split neatly into four pieces2. On later inspection he realized that he could now weigh any mass from one through forty using these four new pieces. How much do each of the new pieces weigh3?
1. 264 – 1
2. Of course it didn’t happen to him. That’s the way he told me the puzzle.
3. 1kg, 3kg, 9kg and 27kg.