Visual Methods

A well-constructed drawing can help to understand or stimulate a mathematical idea. The following puzzles explore some of the visual techniques that can be employed.

We begin by considering representations of the integers, and use the principle that if you count the same thing in two different ways you reach the same total.

We can represent the sum of the first n consecutive integers,

1 + 2 + 3 + … + n

as the quantity of dots in the following figure:

fig1_small.png

By considering an upside down copy of these dots, and by adjoining this copy to the original we are left with a rectangle of n by n+1 dots. The next figure demonstrates this for the case n = 5.
fig2_small.png

So, twice the sum of the first n integers is n(n+1), and therefore:

1 + 2 + 3 + … + n = n(n+1)/2

This is the nth of the so called triangle numbers, normally written Tn, which tend to crop up frequently in combinatorics.

Problem 1: What is 1 + 3 + 5 + … + (2n – 1), the sum of the first n odd numbers?

Next, we consider the dissection and rearrangement of two-dimensional shapes.

A beautiful example is as follows. Given a right angled triangle with sides a, b and c (for the hypotenuse), we observe that we can arrange four copies of it within a square of side a+b in two different ways:

fig3_small.png
The shaded regions in both squares must be equal in area since all we have done is to rearrange the positions of the triangles within the larger square. The shaded region in the first square has area c2, and the shaded regions in the second square have areas a2 and b2. This gives us the famous Pythagorean theorem:a2 + b2 = c2An interesting problem is to cut a cross made up of five squares into four equal parts such that they can be reassembled into a square. The solution is as follows:
fig4_small.png
But how would one arrive at such a solution in the first place?
A common way to approach such problems is to first tile the plane with the shape to be dissected and then tile the plane again with the shape that is our objective.If we can overlay these two tilings in such a way that they match then the required dissection becomes clear. To see what this means we examine the above problem. The cross tiles the plane as follows:
fig5_small.png
By joining the centres of each cross together, we get an overlaid square tiling of the plane:
fig6_small.png
This gives the dissection into four equal parts, but observe that we can slide the two tilings about to get countless other dissections, although not necessarily into equal parts:
fig7_small.png

Problem 2: Cut the following shape, a pentomino, into three pieces (not all equal) that can be rearranged into a square.

fig8_small.png

Then, by considering the more general shape made up of two squares, prove the Pythagorean theorem!

fig9_small.png

Finally, we look at how infinite series can be represented visually.

By starting with a square of side 1, and repeatedly cutting off half of what remains we can arrive at the following infintite dissection:

fig10_small.png
The areas of each piece cut are the geometric series 1/2, 1/4, 1/8, 1/16, … whose nth term is (1/2)n. The dissection demonstrates that the total area of these pieces is 1. That is:1/2 + 1/4 + 1/8 + 1/16 + … = 1If instead we were to repeatedly divide into thirds, we might arrive at the following disection:
fig11_small.png
By shading in pieces corresponding to the geometric series 1/3, 1/9, 1/27, 1/81, … whose nth term is (1/3)n, we see that the shaded and non shaded regions are identical interlocking spirals:
fig12_small.png
The area of each region is therefore 1/2, and so:1/3 + 1/9 + 1/27 + 1/81 + … = 1/2

Problem 3: The following figure is constructed from a square in to which we fit a circle into which we fit a square into which we fit a circle … and so on. What portion of the original square is shaded?

fig13_small.png