Copyright © University of Cambridge. All rights reserved.

I followed a recent email talk-list conversation with interest. The mathematical content does not follow a steady path, because of varying interpretations of the original question. However, the resulting ideas for problems and investigations are worthwhile. A little editing has taken place to transform the informal language of email to publication standard. Some diagrams have also been added
to assist with interpreting the problems. The dialogue began with the following call for assistance:

"Does anyone know of an 'investigation' (other than 'squares on a chessboard') which has the sum of the squares as its solution, and which would be accessible to a group of primary teachers?" (I.T., UK)

The Chessboard problem referred to, is to find out the total number of squares (of all sizes) found on a chessboard, which is an 8 x 8 grid. There are, of course, many ways to go about this. One way is to start by systematically counting from the largest square.

8 | x | 8 | 1 | ||||

7 | x | 7 | 4 | ||||

6 | x | 6 | 9 | and so on, which gives the square numbers. |

The progressive totals give the sequence 1, 5, 14 . . .

"Well, what I have tried with my teacher students is to find a solution to the following problem:

A young girl, Mary, decides to save money during a certain period of time.

She decides that everyday she will put some money in her saving box according to the following pattern: 1 pta (I assume a pound would be too much!!!!) the first day, the second day 2pts, the third day 4pts, and everyday she saves the double of the day before. For how long do you think she will be able to keep on her project? Why?

And how much would she have collected in one month if she managed to follow her intentions?" (N.G., Spain)

This of course is a doubling pattern, set within the context of a real life situation.

"Number of single triangles enclosed by equilateral triangles on isometric paper." (C.A., UK)

This generates the square numbers: 1, 4, 9, 16. If the total number of dots on and inside the triangles are counted, the triangular numbers emerge: 1, 3, 6, 10 . . . (This pattern by the way, fits nicely with the children's book The Very Hungry Caterpillar.)

Now the conversation takes a new turn, with following comment starting off two separate trains of thought.

"Square Pyramid numbers, such as you get in those new Ferrero Rocher boxes (is that the spelling?) would lead to such a solution - but maybe its a bit obvious. Incidentally, I think that 'rectangles on a chessboard' has the sum of cubes as its solution!" (M.S. UK)

"As a chocolate eater, I should perhaps point out that the Ferrero Rocher pyramids have a plastic middle, so that the layers in the pyramid have 1,4,8,12 chocolates in them. (well, my visual aid has plasticene in them as I've eaten the chocs and wrapped up the plasticene in their place.)But of course that's being a bit pedantic.
The idea is fine!

Another suggestion for Ian is the 'steps' investigation with multi-link, where the step shapes build up with successive odd numbers in the layers, so that the flights increase giving the square numbers." (L.J., UK)

"Interestingly, Malcolm, these boxes of chocolate give this impression (of being sum of squares) but are in fact all 'hollowed out' squares. That makes a nice task too - one pyramid number minus another." (T.H., UK)

Now the pyramid idea is developed a little further, bringing out more opportunities for mathematical thinking.

"Sorry not to have been paying attention and so someone may have already suggested this- Think about packing oranges--or stacking tines--in a supermarket--one way of doing this is as a square pyramid with each layer a square and the total given by adding the squares. A nice task is to figure out how to stack a given number of
oranges--cannot begin at the top you see! Aloha." (N. P., Hawaii)

"A very pleasant extension to stacking oranges is to consider the relationship between the volume of the indicative pyramid and the sum of squares, taking cubic oranges of one unit of volume. This, eventually, after some fiddling to account for bits that stick out and bits that stick in, generates the formula for summing squares."
(A.W., UK).

Stacking pyramids with an equilateral triangle base leads to the triangular numbers again! The instigator of the conversation now responds to the rectangles idea.

"Rectangles on a chessboard is even more interesting because the dimensions of the rectangles on, say, the 3x3 board, will be 1x1, 1x2, 1x3, 2x1, 2x2...3x2, 3x3... or, expressed differently, (1+2+3)(1+2+3)... or(1+2+3) squared... or the square of the third triangular number...or 1+8+27, the sum of the first three cubes - thereby
linking triangular, square and cubic numbers in one fell swoop! (I.T., UK)

Now the original question is restated.

"Many thanks for the suggestions. However, what I am looking for is an activity which will generate the numbers 1, 5, 14, 30, 55... without it being obvious that these numbers are the sums of squares. It looks like I'll have to go back to 'squares on a chessboard'... unless you know better..." (I.T., UK)

"Take square spotty paper.

Rotate 45 degrees.

Make a series of simple square diagrams in the obvious way:

I have used this investigation for years. My take on this is to term these "pseudo square numbers". The best starting point is the 5 on a die. (I really wanted to say dice, but decided it best to evade any pedants that might be reading)." (R.T., UK)

The number of dots used in sides of the squares gives: 4, 8, 12, 16.. Counting the number of dots enclosed by each square gives the sequence: 1, 5, 13, 25. So the search for a problem that generates the desired number pattern continues.

With thanks to the members of the Mathematics Education email talklist based at Nottingham, UK.