Counting Squares, Rectangles, Routes and Handshakes

In short: Counting problems are solved by counting in an organised way so nothing is missed or counted twice. Count squares one size at a time, and when $n$ people all shake hands once, there are $\frac{n(n-1)}{2}$ handshakes.

Before you start

• Square numbers: $1,4,9,16,25,36,49,64$.
• Adding a series quickly: $1+2+\cdots +6=21$ (see Shortcuts for Adding a Series).
• Making an organised list or table.

Counting squares in a grid

For a chessboard (8 by 8): $1+4+9+16+25+36+49+64=204$.

Counting rectangles

A rectangle is fixed by choosing 2 of the vertical lines and 2 of the horizontal lines.

Squares are rectangles too, so they are included in this count.

Counting routes on a grid

How many ways from X to Y moving only right or down along the lines?

Handshakes

Seven friends meet, and each shakes hands with every other person once. How many handshakes?

The first person shakes 6 hands. The second has already shaken with the first, so 5 new ones. Then 4, 3, 2, 1, 0.

$$6+5+4+3+2+1=21$$

The same pattern counts games in a round robin (every team plays every other team once) and lines joining dots.

Multiplying choices

If you choose a shirt (3 kinds) and shorts (4 kinds), there are $3\times 4=12$ outfits. "And" means multiply.

Quick summary

• Count squares one size at a time. A 4 by 4 grid has $16+9+4+1=30$.
• A rectangle is two vertical lines and two horizontal lines. Multiply the number of pairs of each.
• Routes moving only right and down: put 1s along the top and left, then add the number above and to the left.
• Handshakes among $n$ people: $\frac{n(n-1)}{2}$.
• Making one choice and another: multiply the number of options.

Quiz