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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.

When Will You Pay Me? Say the Bells of Old Bailey

Age 11 to 14 Challenge Level:

An excellent introduction to this sort of bellringing can be found in Ringing Elementary Minor Methods on Handbells by Frank H. King (the Cambridge University Bellringer).You may also like to take a look at Ding Dong Bell, an article by Toni Beardon.

This problem is the third of three related problems. The first problem is Oranges and Lemons, Say the Bells of St Clement's , and the second is You Owe Me Five Farthings, Say the Bells of St Martin's .