In this task you will follow a strategy to solve a mathematical puzzle.
The Problem
How many squares are there in a chessboard?
(The black squares aren't coloured in to help spot "bigger" squares)
There are way more than 64!
The real issue is, there are so many squares, it's hard to make sure you got them all.
Task 1: Try Something Smaller
Look at the diagram below.
(a) How many squares are there that are of size 1 × 1?
(b) How many squares are there that are of size 2 × 2?
(c) How many squares are there of other sizes?
(d) How many squares are there in total?
(e) What methods or aids can you come up with that can help you keep track of all possibilities?
(f) How can you be sure you have found all the squares in the diagram?
Task 2: Tackle The Problem
Back to the 8 × 8 grid.
(a) How many different sizes of square are in the diagram?
(b) Construct a table listing the size of a square and how many squares of that size are in the diagram.
(c) What is the total number of squares in the diagram?
(d) How can you be sure you have found all the squares?
Task 3: Find A Rule
(a) Is there a pattern or formula you can use to count squares in a square-grid diagram?
(b) How many squares would be in a diagram that was a 12 × 12 grid of squares?
(c) How many squares would be in a diagram that was a 20 × 20 grid of squares?
Task 4: Extend The Problem
(a) How many squares are in a 2 × 4 grid of squares?
(b) How many squares are in a 3 × 5 grid of squares?
(c) How many squares are in a 4 × 6 grid of squares?
(d) Is there a pattern that you can use to predict the number of squares in a grid that has two extra columns than rows?
(e) How many squares would be in a diagram that was a 10 × 12 grid of squares?
(f) How many squares would be in a diagram that was a 20 × 22 grid of squares?
(g) Is there a pattern that you can use to predict the number of squares in grids of any size?
(h) How many squares would be in a diagram that was a 10 × 20 grid of squares?
(i) How many squares would be in a diagram that was a 5 × 30 grid of squares?
Task 5: Going Even Further
Look at the diagram below:
(a) How many different kinds of rectangles fit into the diagram?
(b) Make a table showing the different types of rectangles and how many of each there are in the diagram.
(c) How many rectangles are there in total?
(d) Is there a pattern or rule that you can use to predict the number of rectangles in grids of any size?
