### 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.

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

##### Stage: 3 Challenge Level:

We received a very clear explanation from Vicky outlining how she works out when she should ring her bells. This is what she wrote:

On each row, I counted "1, 2, 3, 4", emphasizing the places where my bells needed to ring.

When it was time for my bells to change order, I tried to remember that (it didn't happen very often, so it was quite easy to remember).

For example, when I was ringing bells 1 and 2 when there were four bells, it went like this:
1 , 2 , 3, 4
1 , 2 , 3, 4
(other way round) 1 , 2 , 3, 4
1 , 2, 3 , 4
1, 2 , 3, 4
1, 2, 3 , 4
(other way round) 1, 2, 3 , 4
1, 2 , 3, 4
1 , 2, 3 , 4
1 , 2 , 3, 4.

Instead of trying to remember long lists of numbers, I instead remembered the patterns that my bells made. That way, on each row I 'only' had to work out where my bells were going to be on the next row.

I found the picture really useful for remembering this, but I also thought about the pattern too. During the main part of the pattern, my bells were two apart, moving one place to the left or one place to the right on each row. (It's not very obvious that this is the main part of the pattern when there are only four bells, but it's clearer when there are six or eight bells. Look at the picture and you'll see what I mean.) When one bell reached the end of the row, it waited there for the other one to catch up, and then they swapped places. Then one stayed still while the other moved out so that they were two apart again, and they did the main part of the pattern. This made it quite easy to remember (with a bit of practice!).

The handy thing about this is that I didn't have to learn a new pattern when I wanted to ring bells 3 and 4, because they just do the same thing (starting in a slightly different place), so it was easy to work out how their pattern went.

A similar thing worked for six bells, where bells 1 and 2 and bells 5 and 6 ring the same pattern as described above. However, bells 3 and 4 do something slightly different. Once I'd got the hang of ringing bells 1 and 2, however, it wasn't too hard to learn a new pattern. This time it was symmetrical, and the bells moved further apart or closer together on each row. The hardest bit was where one bell was in place 6 and the other in place 1: I kept forgetting that I was effectively ringing two consecutive bells!

When there are eight bells, bells 1 and 2 and bells 7 and 8 do the same thing as described above. Bells 3 and 4 and bells 5 and 6 do something slightly different, but again the same way of thinking works.

Apparently there are 16 bells at the Bull Ring in Birmingham , but I think that similar ideas would work there. I'm sure that they ring more complicated changes, but they probably still think about following the lines in the patterns in the same way.

Thank you Vicky.