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Picture a Pyramid ...

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Lydia from Rickling C of E Primary made a fantastic start to this problem. She told us:

I took the number of the layer and squared it making layer one have 1 cube, layer two have 4 cubes etc.
Starting from the North West corner going left to right from the top, I numbered the cubes so it looked like this:

1
layer 1

2 3
4 5
layer 2

6 7 8
9 10 11
12 13 14
layer 3

Etc ...
I the answered the questions by looking at the numbers eg; cube 10 is vertically under cube 1 because cube 10 is in the middle of that layer.

Well done Lydia - this is a very helpful way to picture it. Are there any other cubes directly underneath cube 1?

Tommy from Brede School also had a really good go at this problem. Tommy agreed with Lydia that cube 10 is directly beneath cube 1, but he didn't look to see whether there were any more cubes underneath them either. Tommy went on to say:

The cube that is under number 4 is 24
The cube that is under number 8 is 39

I wonder whether there are any other cubes underneath these? Have a think!

Stepping up in a straight line is 1 &13 (Or perhaps 13 then 1? And what number is the very first step?)
Stepping up on corner steps are 86, 51, 27, 12, 4 & 1

Thank you very much for your solution, Tommy. I wonder how you worked out these answers?

Did anyone try the very last part where the cubes were numbered differently? If you have a go at this, then do let us know.