Plummy Pyramids

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

Sometimes these are in nice square arrangements, but, more often, fruit boxes are rectangular, rather than square…

Typically, pyramids of large fruit may have 6 by 4 oranges or peaches on the bottom layer, as seen above, while pyramids of small fruit may have 10 by 7 plums or apricots on the bottom layer.  The top layer will be a short row of fruit, in the above case a row of 3 oranges.

Problem 1

How many fruit would a ’10 by 7’ pyramid have on the top row?

How many fruit altogether?

How many ‘invisible’ fruit inside the pyramid?

Problem 2

Work out how many fruit will be in each of these pyramids:

(‘10 by 4’ means the base layer of plums contains 10 rows of 4 plums)

A         10 by 4           B         9 by 5              C         8 by 6              D         7 by 7

# Now use a Spreadsheet to find how many fruit can be piled up in pyramids using the base of each of these boxes.

A         10 by 8           B         12 by 11                     C         15 by 9                       D         17 by 14

# Problem 4

The formula for the total number of plums depends on both the width (w) and the length (L) of the base layer.

If I tell you that the formula is of the form     P = w x (…?) x (3L – w + 1) ¸ 6,

can you finish it off? Use your previous results…

What happens if you make both the width and the length equal to n?

(See “Peachy Pyramids”)

Problem 5

Use your Spreadsheet to find what base layers you would need to start with in order to make the following total numbers of apricots into a nice neat pyramid.

A         50        B         70                    C         145                 D         175

You’re now ready to work on your Salesman/woman’s voice and patter…

“Come on then – lovely lot of plums ‘ere now – straight from Egypt, fresh as an ibis. Come on then – who’ll ‘ave 217 plums then? Beautiful bargains – step this way”

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