What are the missing numbers in the pyramids?
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?
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.
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?