The answer isn't 64 — discover the sum of squares hiding inside a chessboard
/ 24
♟️ The Challenge
Topics: systematic counting, sum of squares, algebraic proof, generalisation
Question: How many squares of all sizes can be found on a standard 8×8 chessboard?
Most people say 64 — counting only the 1×1 squares. But the board also contains 2×2 squares, 3×3 squares… all the way up to the one 8×8 square.
Count every square of every size. What is the total?
Key insight: On an 8×8 board, a k×k square can be placed with its top-left corner at any of (9−k) positions horizontally and (9−k) positions vertically.
So the number of k×k squares = (9−k)²
Total = Σ(9−k)² for k=1 to 8 = 8² + 7² + 6² + … + 1² = n(n+1)(2n+1)/6 where n=8
Sum of squares formula: 1² + 2² + 3² + … + n² = n(n+1)(2n+1) / 6
For n = 8: 8 × 9 × 17 / 6 = 1224 / 6 = 204
📝 Questions
1
On a 4×4 board (simpler version), count the number of 1×1 squares, 2×2 squares, 3×3 squares, and 4×4 squares. Write each count. 4 marks
Square size
1×1
2×2
3×3
4×4
Total
Count
2
On an 8×8 board, a 1×1 square can have its top-left corner in any of 8 positions horizontally and 8 positions vertically. How many 1×1 squares are there? Now count 2×2 squares (top-left can be in 7 positions each way). How many? 2 marks
3
Write a formula for the number of k×k squares on an 8×8 board. Use it to complete the table. 4 marks
k (size)
1
2
3
4
5
6
7
8
Count
64
49
4
Add up all the counts from your table. What is the total number of squares of all sizes on an 8×8 chessboard? 2 marks
5
The total equals 1² + 2² + 3² + … + 8². Use the sum of squares formula n(n+1)(2n+1)/6 with n = 8 to verify your answer. 3 marks
6
Using the formula, how many squares of all sizes are on a 10×10 board? A 5×5 board? 3 marks
7
Extension: On an 8×8 board, how many rectangles (of all sizes, including squares) are there in total? Hint: a rectangle is defined by choosing 2 of the 9 vertical lines and 2 of the 9 horizontal lines. 6 marks