Peachy Pyramids

Classy greengrocers and fruiterers often pile up their fruit and veg in beautiful pyramid arrangements.

Often these are in nice square arrangements, with each layer fitting into the ‘gaps ‘ in the layer below.

Hence each layer is a square, but one fruit shorter on each side than the layer below.

Problem 1

How many layers are there in the ‘peachy’ pyramid shown above?

How many peaches would the pyramid contain altogether?

How many ‘invisible’ fruit inside the pyramid?

Problem 2

Use a Spreadsheet to complete a table showing the Total numbers of fruit, and the number of Hidden fruit in each square fruity pyramid from 1 by 1 up to 20 by 20.

Problem 3

The formula for the total number of peaches depends on the width of the base layer, which we can call n.

If I tell you that the formula is of the form     P = n x (n + 1) x ( … ?) / 6,

can you finish it off?

# Problem 4

Can you modify your Spreadsheet to give the answers for pyramids built up from a triangular base layer? Or, even better, find the formula…!

(As a hint, the formula for the Triangle numbers on each layer, as opposed to Square numbers, is

n x (n + 1) / 2, as opposed to n ^ 2.)

Open the File as a Word Document