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What are the missing numbers in the pyramids?

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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.


Age 11 to 14 Challenge Level:

On a certain island there are 12 green, 15 brown and 18 yellow chameleons. Whenever two chameleons of different colours meet they always change colour to the third colour (e.g. a brown and a yellow would both change to green when they met). This is the only time they change colour. Describe the shortest sequence of meetings in which all the chameleons change to green.

Now suppose there are 13 green, 15 brown and 17 yellow chameleons and they change colour in exactly the same circumstances. Is it possible now for all the chameleons eventually to be the same colour?