### 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.

# Oranges and Lemons, Say the Bells of St Clement's

##### Stage: 3 Challenge Level:

We received correct solutions from Alex, Alice, George, Nell and Tom from Gorseland School, Oliver, Edward, Alex and Paul from Dartford Grammar School, Katie from Bathurst West PS, Adam, Annie, Vijay, Louise, Jude, Aaron and Mary from St. Mary Star of The Sea, Ben from Victoria College, Bethany, Alicia and Grace from Willaston Primary School, Samantha and Frances from St. Alban's Catholic Primary School and Max and William from Brenchley and Matfield School. Well done to you all.

Alex, Alice, George, Nell and Tom sent in this diagram showing the paths of the six bells:

The pattern of the paths for eight bells was sent by Edward:

Max and William also sent us a solution for twelve bells!

1 2 3 4 5 6 7 8 9 10 11 12
2 1 4 3 6 5 8 7 10 9 12 11
2 4 1 6 3 8 5 10 7 12 9 11
4 2 6 1 8 3 10 5 12 7 11 9
4 6 2 8 1 10 3 12 5 11 7 9
6 4 8 2 10 1 12 3 11 5 9 7
6 8 4 10 2 12 1 11 3 9 5 7
8 6 10 4 12 2 11 1 9 3 7 5
8 10 6 12 4 11 2 9 1 7 3 5
10 8 12 6 11 4 9 2 7 1 5 3
10 12 8 11 6 9 4 7 2 5 1 3
12 10 11 8 9 6 7 4 5 2 3 1
12 11 10 9 8 7 6 5 4 3 2 1
11 12 9 10 7 8 5 6 3 4 1 2
11 9 12 7 10 5 8 3 6 1 4 2
9 11 7 12 5 10 3 8 1 6 2 4
9 7 11 5 12 3 10 1 8 2 6 4
7 9 5 11 3 12 1 10 2 8 4 6
7 5 9 3 11 1 12 2 10 4 8 6
5 7 3 9 1 11 2 12 4 10 6 8
5 3 7 1 9 2 11 4 12 6 10 8
3 5 1 7 2 9 4 11 6 12 8 10
3 1 5 2 7 4 9 6 11 8 12 10
1 3 2 5 4 7 6 9 8 11 10 12
1 2 3 4 5 6 7 8 9 10 11 12

They even tried to find a solution for fifteen bells, but because it is an odd number of bells they weren't able to stick to the rule that on alternate rows we swap all of the bells (in pairs) and on the remaining rows, we fix the bells on either end and swap the rest.