Published February 2011.
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:
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.
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.
Now the pyramid idea is developed a little further, bringing out more opportunities for mathematical thinking.
Stacking pyramids with an equilateral triangle base leads to the triangular numbers again! The instigator of the conversation now responds to the rectangles idea.
Now the original question is restated.
With thanks to the members of the Mathematics Education email talklist based at Nottingham, UK.