How many squares on a chessboard?

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Presentation transcript:

How many squares on a chessboard? 1 x 1 = 64 2 x 2 = 49 3 x 3 = 36 4 x 4 = 25 5 x 5 = 16 6 x 6 = 9 7 x 7 = 4 8 x 8 = 1 Clue 1 There isn’t a clue 2. 204 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 = 204

There is also a formula for adding successive square numbers. You may be familiar with the formula that adds up successive whole numbers. How is it Derived?. There is also a formula for adding successive square numbers. Check that it works for the chessboard problem. How is it Derived?. Research a formula that adds up successive cube numbers.

A Knights Tour

A Knights Tour of a 6 x 6 Chessboard 1 A Knights Tour of a 6 x 6 Chessboard 16 22 21 2 15 23 17 6 9 4 11 20 3 18 7 14 8 5 10 12 19 13

A Knights Tour of a 6 x 6 Chessboard 1 A Knights Tour of a 6 x 6 Chessboard 24 11 32 18 10 33 2 17 4 31 23 16 25 12 19 26 9 34 3 30 5 15 7 13 22 28 20 8 27 14 21 6 29

A Knights Tour of a 6 x 6 Chessboard 1 A Knights Tour of a 6 x 6 Chessboard 4 7 32 11 18 6 33 2 17 8 31 16 3 5 10 19 12 34 36 23 30 27 9 15 28 13 20 22 25 26 35 14 21 24 29

A Knights Tour of an 8 x 8 Chessboard 2 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 39 60 64 63 56 59 57 61 62 58 55 A Knights Tour of an 8 x 8 Chessboard De Moivre’s Solution

A Knights Tour of an 8 x 8 Chessboard Euler’s Magic Square Solution 2 1 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 23 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 43 44 45 46 47 48 50 51 52 53 54 60 64 63 56 59 57 61 62 58 55 49 24 39 34 36 22 11 260 What’s the magic number? 260

A Knights Tour of an 8 x 8 Chessboard Euler’s re-entrant half-board Solution 2 1 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 23 25 26 27 28 29 30 31 32 33 35 37 38 40 41 42 43 44 45 46 47 48 50 51 52 53 54 60 64 63 56 59 57 61 62 58 55 49 24 39 34 36 22 11