Thursday, May 15, 2014

Happy Mercuralia!

Today is Mercuralia, the ancient Roman festival of the god Mercury. Mercury, like his Greek cousin Hermes, is the god of the Mind: mathematics, writing, fast-talk, inventors, commerce, travel, crossroads, dreams, etc. Edith Hamilton calls Mercury the god of "mathematicians, thieves and other ne'erdowells"

In honor of the festival of the god of puzzles, here's a favorite puzzle of mine. One thing I like about it is that is comes with a legend. Here's the story:

Far off, in the misty Himalayas, is a secret and ancient monastery. Built into the side of a mountain, it appears to have been carved out of the rock by the hand of the gods. In the central courtyard, there stand three massive and antediluvian stone pillars, as tall as the temple's peaked roof, stretching upward to the sky like great granite trees. About the pillars are golden disks, carved with the holy symbol of the Sun, Moon, the Stars, and the Planets. There are 64 disks in total, of various sizes. The largest is as far across as 3 men, laid head to foot, and the smallest is the size of a teacup.

When the world was new, say the monks, the gods arranged all the disks on the easternmost pole, with the largest on the bottom and the smallest on the top. Every day, the gods commanded, the monks must work together to move one disk. By holy decree, a larger disk can never be placed on top of a smaller one. This teaches us that the strong must always support the weak. And so, every day, the monks come together to move one of the disks. The young strong monks struggle together, with elaborate systems of ropes and pulleys, to move the heaviest of the golden, coins. This teaches us the value of dedication and team work. The older, wiser monks spend all their time thinking and calculating, deciding which disk to put where. This teaches, say the monks, the value of careful planning, and collaborating to solve hard problems.
It is the movement of the disks, say the monks, that drives all progress in the world. As the disks move slowly from the east to the west, people are born, civilizations rise, humans learn and grow and prosper. We create art, and discover truths, and invent medicines. Sometimes, however, disks must temporarily move backwards, and then there is war and famine and plague and death. When the disks are all on the western pillar, the world will be perfected, and all beings will attain nirvana.
This ancient order of monks, secret and hidden, has been moving the disks as long as anyone can remember, thousands and thousands of years, perhaps, as the monks themselves claim, since the very day he world was made.

The First Puzzle: Play the game with fewer disks. You can try it here: The drop down box in the center bottom lets you choose how many disks to use. Three is too easy and seven is probably too hard. Press the green button to start over.

The Second Puzzle: If the monks have been moving the disks, one per day, how long until we all reach nirvana? (Assume the monks play perfectly, that is, they solve the puzzle in the fastest way possible,and never make a mistake.) Hint: try playing the game with various numbers of disks, and see if you can develop a strategy. Then, examine your strategy to get a formula for the number of moves it takes to solve.

If you'd like to learn more about the math behind this puzzle (or you want spoilers) google "The Towers of Hanoi".

Happy Mercuralia!

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