Towers of HanoiAuthor(s): John SlaterSource: Mathematics in School, Vol. 23, No. 2 (Mar., 1994), pp. 28-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215094 .

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TO~ERS

OF

HANOI

by John Slater Beechen Cliff School, Bath

The Activity The Towers of Hanoi (or Brahma) is described in two books of mathematical puzzles which sit on my shelves1',2. The following description appears in an earlier book by Ball'.

In the great temple at 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 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 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 thunder-clap the world will vanish.

Some time ago I used this as the starting point of an investigation undertaken by two fourth year classes of moderate "Higher" level ability in successive years. On the second occasion, the class worked on the problem in the first term of the year, whereas the older class attempted the exercise later in the course. I was able to make some assessment of my own success in conducting investigations, and noted that the younger group seemed to achieve more because of the experience I gained earlier on the same investigation.

Within our department we have sometimes expressed concern at the ease with which ideas are transferred within a class when pupils are attempting work which is to be assessed. But confidence grows -

teachers become adept at picking up the source of ideas. On the second occasion, I positively encouraged discussion at various points during the investigation, and noted the source of ideas as work progressed.

Why is it my notes never have that such carefully researched look that appear in some articles? Perhaps because I find it difficult to spend time writing up in detail what pupils have said to me before I forget it. I know I prefer to get stuck into the problems myself. And on this occasion, that is exactly what happened.

Classroom Activity So where did the class start? We spent a quarter of an hour or so with card and scissors cutting out a set of circles of increasing size after introducing the problem. With an initial description of the only two rules:

(a) that only one disc may be moved at a time, and

(b) that no disc may be placed on top of a smaller disc, pupils soon found systems of moving a set of discs from one pile to another.

Many pupils find it difficult to write down what they are doing at an early stage, even in a very rough form, so I laid some emphasis on the importance of keeping a record of progress so that it would be written up effectively. Then it was every one for themselves for the remainder of an hour's lesson and some homework.

Within the first half hour several pupils had devel- oped shorthand systems for recording moves. This was the first to emerge where the three poles are labelled A, B and C, and the discs are numbered, 1 being the smallest:

1B - 2C - 1C- 3B 1A -- 2B 1B

I call this the "Al" notation. What did it mean, I asked? Could the inventor then tell me where disc 2 had moved from when it was placed on pole B (the second to last move)?

Another had produced something similar, but held redundant information compared to the first:

A1B - A2C - B1C - A3B - C1A - C2B - A1 B

Yet another had produced a more visual notation - the * replaces a red dot which the pupil had used to "mark the previous position of a moved disc", so I will call it the "red dot table".

28 Mathematics in School, March 1994

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Number Number Pole Pole Pole of discs of moves 1 2 3

3 Start 3,2,1 1 3,2 * 1 2 3* 2 1 3 3 2,1 4 * 2,1 3 5 1 2* 3 6 1 * 3,2 7 * 3,2,1

Developing the Activity Both types of notation proved to be fruitful in developing ideas. On arrival at the next lesson, I mixed up the class, putting pupils from different areas in the classroom together at one desk, and asked them to spend some twenty minutes probing, pooling and testing ideas. What had developed?

Two fundamental ideas emerged as we went back into whole class discussion:

(a) the largest disc moves only once - on the "central" move, and

(b) every set of moves before the central move is mirrored in reverse with two of the poles exchanging their roles.

Anxious not to let too much out of the bag straight away, they were all directed back to their own work, and I watched as I privately ruminated further.

What if ...? We pursued a "what if?" discussion before they departed for yet another homework.

The third lesson. "What if?" was bearing fruit. More poles had been my suggestion. How about having two of each size of disc? How about colouring such discs so no two similar colours may come into contact? All sorts of wonderful possibilities developed, far beyond the likelihood of development within the time-scale I had envisaged. Using more poles had been tackled in the investigation conducted by the older group, and I had been forced under pressure of marking time to put a comment on one piece of work summing up the moves of up to 9 discs on 4 poles:

"plausible - is it true?" Here it is (with differences in brackets):

Number Moves with Moves with of discs 3 poles 4 poles

1 1 (+2) 1 (+2) 2 3 (+4) 3 (+2) 3 7 (+8) 5 (+4) 4 15 etc. 9 (+4) 5 31 13 ( + 6) 6 63 19 (+6) 7 127 25 (+8) 8 255 33 (+8?) 9 511 41?

I decided the value 19 for 6 discs using 4 poles was critical. A physical test with some card discs yielded a minimum of 17 moves - once! I couldn't repeat it!

Formulas started to appear. Number of moves for n discs=2n-1 emerged from one corner. Matthew had "seen a computer program" and knew the answer. This was like a red rag to a bull! I knew there must be

a way of doing this for more poles, but that is for later. Interestingly, his final work was by no means the best. More impressive was a simpler formula:

number of moves for n discs= 2n - 1 if there are at least n+l poles. Systems for working out the differences were another common feature of predicting the minimum number of moves for larger numbers of discs.

On Reflection In retrospect, I can split my pupils into two rough groups in terms of problem solving.

The majority see a sequence of numbers, spot the system or formula and predict what happens for larger values. This prediction "must be true" because the formula works.

A minority realise that it is not the ability to see the formula or system which is important, but to be able to relate this formula or system back to the concrete example. Thus the inventor of the "red dot table" had seen what he called symmetry in his table, although it was "twisted". Some of those who had developed the "Al" type notation noticed that the smallest disc moved most often, the next size half as much, and so on. Others even noticed patterns in the sequence of poles used, so were able to give rules about WHERE to move the next disc, and not just which disc to use. F*

References 1. Northrop, E. P. (1945) Riddles in Mathematics, 2. Gardner, M. (1961) Mathematics Puzzles and Diversions, 3. Ball, W. R. R. (1974) Mathematical Recreations and Essays,

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