29 Mar '06 14:431 edit

A colleague of mine's son was given the following exercise to do in class, as an exercise in subtracting numbers.

Start with a largish piece of paper and write four random numbers in a square:

2 17

7 9

Then put the positive difference of each corner in the centre of each side, forming a diamond thus:

2..15..17

5.........8

7....2...9

Do the same thing with the diamond thereby forming another, smaller square, and keep going:

2..15..17

..10...7..

5.........8

...3....6..

7....2....9

What happens? The kids were supposed to notice that eventually the process ends up with four zeros.

The questions asked of me were "Does it always end no matter what numbers you start with? And if so, what is the maximum number of iterations it can take"

It appears that eight iterations is the maximum as per the example above, but I couldn't come up with anything like a concrete proof. Anybody got any thoughts, counter examples, etc?

EDIT $%^£ing spacing!

Start with a largish piece of paper and write four random numbers in a square:

2 17

7 9

Then put the positive difference of each corner in the centre of each side, forming a diamond thus:

2..15..17

5.........8

7....2...9

Do the same thing with the diamond thereby forming another, smaller square, and keep going:

2..15..17

..10...7..

5.........8

...3....6..

7....2....9

What happens? The kids were supposed to notice that eventually the process ends up with four zeros.

The questions asked of me were "Does it always end no matter what numbers you start with? And if so, what is the maximum number of iterations it can take"

It appears that eight iterations is the maximum as per the example above, but I couldn't come up with anything like a concrete proof. Anybody got any thoughts, counter examples, etc?

EDIT $%^£ing spacing!