Learning Session 2 – Chessboard rectangles
1. How many rectangles in a chessboard?
How many different sizes of rectangles are there in a chessboard with n
squares on a side?
n
r
1
1
2
3
3
6
4
10
5
15
…
For an n by n chessboard, the number of sizes of rectangles is the nth triangle number
r = n(n + 1)/2
2.
r
r
r
Rectangles of 2 squares on a chessboard.
1
0
4×0
2
4
4×1
3
12
4×3
4
24
4×6
5
40
4 × 10
…
For an n by n chessboard, the number of sizes of rectangles is 4 times the (n – 1)th triangle number
r = 4n(n – 1)/2
3. Students will have to start with a smaller problem and be systematic about counting.
They might start by counting squares, then 1 by 2 rectangles, 1 by 3, 2 by 3, etc, think about vertical
and horizontal rectangles then look for patterns in the answers. Having worked their way through
the previous problems, students might recognise the pattern as the squares of triangle numbers.
n
r
r
1
1
1
2
4 +1+4
9
3
2(6 + 3 + 2) + 1+ 4 + 9
36
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