Pyramids

What are the missing numbers in the pyramids?

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?

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.

You Owe Me Five Farthings, Say the Bells of St Martin's

Stage: 3 Challenge Level:

We received good solutions from Gareth, Ed and Omer from Dartford Grammar School, Harry from Culford School and Alex, Alice, George, Nell and Tom from Gorseland School. Thank you all.

Harry explained how he knows when it is his turn to ring:

An easy way to remember when to ring is to notice a pattern.
e.g. 1, 1, 2, 3, 4, 4, 3, 2, 1, 1 is the exact pattern that bell 1 takes.
Your turn is first, then first again, then second, then third, and so on (in each set of four).

After every ring, bell 1 initially rises up further to the back.
1234
1234
2143
2413
4231 ...
For bell 1 the pattern is: same, more, more, more, same and so on.
For bell 2 the pattern is: same, less, same, more, more, and so on.
For bell 3 the pattern is: same, more, same, less, less and so on.
For bell 4 the pattern is: same, less, less, less, same and so on.
From these data an easy pattern to remember for bell 1 is 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4...

Alex, Alice, George, Nell and Tom from Gorseland School sent the following explanation:

There is a certain pattern each bell follows.
Bell 1 rings first on the first sequence of four rings, second on the second sequence, third on the third sequence and fourth on the fourth sequence.
Then, on the fifth sequence, the bell rings fourth again, there being four bells.
On the sixth, Bell 1 rings third again, second on the seventh sequence, first on the eighth sequence, first on the ninth sequence, second on the tenth sequence, and so on.

Each bell follows this pattern but starts in a different place.

Ken, a volunteer (and bellringer) helping at their school wrote:

After half an hour of intense discussion about the problem, and the various ways they might solve it, one of the children asked "Are we not doing any maths this morning?"
They then wanted to stay in during break to see if they could actually ring the bells - and did so quite well.

We are delighted you enjoyed the problem.