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?
Use your previous results, and a new column in your Spreadsheet to try out your guesses.
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