# Tower of Hanoi

The tower of Hanoi is a game that works on multiple levels. It is a challenging game that test the agility and organization skills of the player. It is also a game mathematicians would love since the game is an excellent illustration of math concepts such as mathematical induction and exponential growth. It is also a concrete illustration of a recursive algorithm.

The game of tower of Hanoi involves moving disks from one rod into another rod. The following is a tower of Hanoi game with 8 disks and three rods.

The goal of the game is to move all the disks from the left most rod to the right most rod, one disk at a time. Only the uppermost disk on a rod can be moved. In addition, you can only place a smaller disk on top of a larger disk.

Tower of Hanoi sets such as the one shown above are available from Amazon. A home made tower of Hanoi set can also be created. For example, use paper to mark three spots (to serve as rods). Then stack books of varying sizes in one spot and proceed to move the books to another spot according to the rules described above. Kitchen plates can also be used in place of books.

There are also many online versions of the game. Tow examples: version from Math is Fun (from 3-disk to 8-disk game) and 3-disk game to 9-disk game are available in this version. Both sites are easy to use. I prefer the Math is Fun version since the disks are in different colors. Of course, there are many others to choose from (simply Google tower of Hanoi game).

Obviously, the more disks there are in the game, the more difficult it is to successfully to transfer the disks. It is possible that the player may make more moves than necessary if the player is not organized or gets lost.

A 3-disk game can be played in 7 moves and no less than 7 moves. A 4-disk game can be played in a minimum of 15 moves. For a player who gets lost may end up taking more than 15 moves in a 4-disk game. Any player in the know can finish the 4-disk game in 15 moves. The 5-disk game can be played in 31 moves. For 6-disk games, 63 moves. For 7-disk game, 127 moves. To see these for yourself, explore the game using a home made set or play online. The game is also discussed here in a companion blog.

Notice that whenever an additional disk is added to the game, the minimum number of moves is doubled, e.g. from 7 moves to 15 moves (from 3 disks to 4 disks), from 15 to 31 (from 4 disks to 5 disks) and so on. In general, the $n$-disk game requires a minimum of $2^n-1$ moves. Thus the tower of Hanoi is a concrete example of an illustration of exponential growth – increasing the size of the game by one disk doubles the time required to play the game.

In general exponential growth is a phenomenon such that increasing the input by one unit increases the output by a constant multiple (e.g. doubling, tripling, or multiplying with other constant). In contrast, linear growth (or growing linearly) means that increasing the input by one unit increases the output by a constant but as an additive amount.

The exponential growth is even easier to see if the moves are converted into time. Assume that it takes one second to move a disk. It would take 63 seconds to play the 6-disk game, roughly one minute. It would take 127 seconds to play the 7-disk game, roughly 2 minutes. In that two minutes, the play would need to know exactly what the moves should be. Otherwise it would be easy to make a mistake and hence taking more moves than necessary. So converting the moves to seconds further illustrates the exponential growth inherent in the tower of Hanoi game.

A more subtle aspect of the tower of Hanoi game is that in order to play it successfully (i.e. in the minimum number of moves), the game must be played recursively. Take the 4-disk game as example. Imagine that the 4 disks are at first in the left rod. The goal is to move them to the right rod (the destination rod). The rod in the middle is the intermediate rod. The strategy is to move the first 3 disk to the middle rod. Then move the 4th disk (the largest disk) from the left rod to the right rod. The remaining moves will be to move the three disks in the middle rod to the right rod.

With $n$ disks, move the top $n-1$ disks into the intermediate rod (by following the rules of course). Then move the largest disk in the starting rod into the destination rod. To finish off the game, move the $n-1$ disks in the intermediate rod into the destination rod. So the $n$-disk game is executed by playing two $(n-1)$-disk games with the move of the largest disk in between. So the tower of Hanoi is a great introduction to a recursive algorithm. The tower of Hanoi game would be a great computer programming exercise.

Because of the recursive nature of the game, it would be a challenge to keep track of the moves when the number of disks is large. In a 4-disk game, you would play 3-disk games twice with one move of the largest disk in between. This can be managed with ease after some minimal practice. Say, you want to play the 8-disk game, you would need to play 7-disk game twice with one move of the largest disk in between. For each of the two 7-disk games, you would need to play 6-disk game twice with one more move in between. That would mean four 6-disk games. Then in each of the 6-disk game, you need to play 5-disk game twice with one more move in between. The recursion can get complicated fast! It will be helpful to use diagrams to keep track of all the sub games that are required in the recursive algorithm. This is discussed here in a companion blog.

Now that we know adding one disk to the game of tower of Hanoi doubles the number of moves, hence doubling the time it takes to play. What about doubling the number of disks?

The 8-disk game only requires a minimum of 255 moves (about 4 minutes with one second per move). The 16-disk game would require 65,535 moves, over 1,000 minutes (assuming one second per move) or over 18 hours! The following shows a 32-disk tower of Hanoi set, which is located in a museum in Mexico.

A 32-disk game would require $2^{32}-1$ moves, which is 4,294,967,295. Assuming one second per move, this would be over 136 years! If the workers in the museum is required to move the disks from one rod to another rod by following the rules of the game, that’s would be job security!

The game of Tower of Hanoi is a deceptively simple game. Yet the effect of doubling the number of disks is very dramatic. What about doubling the number of disks to 64, twice as many disks as the one shown above? The following is an interesting tale of the origin of the game of Tower of Hanoi [1].

In the Temple of Benares, beneath the dome which marks the centre of the world, rests a brass-plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee. On one of these needles, at the 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 Bramah! Day and night unceasingly the priests transfer the discs from one diamond needle to another according to the fixed and immutable laws of Bramah, which require that the priest 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 below it. When the sixty-four discs shall have been thus transferred from the needle on which at the creation God placed them to one of the other needles, tower, temple and Brahmins alike will crumble into dust, and with a thunderclap the world will vanish.

The game of the tower of Hanoi was invented by the French mathematician Édouard Lucas in 1883. A year later, an author called Henri de Parville told of the above interesting tale about the origin of the tower of Hanoi.

It is not known whether Lucas, the inventor of the game, invented this legend or was inspired by it. One thing is clear. The legend accurately describes the enormity of the 64-disk game of the tower of Hanoi.

The least number of moves that are required to play the 64-disk game is $2^{64}-1$, which is 18,446,744,073,709,551,615, when converted to years would be 585 billion years (again, assuming one second per disk). In contrast, the age of the universe is believed to be 13.82 billion years. The age of the sun is believed to be 4.6 billion years. The remaining lifetime of the sun is believed to be around 5 billion years. So by the time the sun dies out the game is still not finished!

Back to the question about what happens when the number of disks is doubled. For the 8-disk game, the number of moves is 255. For the 16-disk game, the number of moves is 65,535. Note that the square of 255 is 65,025. So doubling the number of disks has the effect of squaring the number of moves. This is another demonstration of exponential growth.

Reference

1. Hinz Andreas M., Kla. Dzar Sandi, Milutinovic Uros, Ciril Petr, The Tower of Hanoi – Myths and Maths, Springer Basel, Heidelberg, New York, Dordrecht, London, 2013.

$\copyright$ 2017 – Dan Ma

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