In brief
On the chessboard, a child works on locating by coordinates (each square has a letter and a number), geometry (ranks, files, diagonals), parity (the bishop never changes colour), distances, arithmetic (piece values) and counting. These notions match the primary and secondary school curricula, which makes chess and maths in play a natural vehicle for the classroom.
Chess is not a maths lesson in disguise, but it is very close to one. A child who plays handles, without knowing it, concepts that the maths curriculum asks them to master. The adult's role is to make these links visible. The exercises below have all been checked.
- Locating on a grid
- The skill of designating a square on a grid with two pieces of information, its column and its row. On the chessboard, the files are labelled a to h and the ranks 1 to 8: the square e4 is at the intersection of file e and rank 4.
Chess and maths in play: overview
| Notion | What you see on the chessboard | Level |
|---|---|---|
| Locating | Square coordinates (a1 to h8) | Cycle 2 |
| Geometry | Ranks, files, diagonals, symmetry of the starting position | Cycles 2 and 3 |
| Parity | The bishop stays on its colour, the knight changes colour at every jump | Cycle 3, secondary school |
| Distances | Number of king moves, the rule of the square | Cycle 3, secondary school |
| Arithmetic | Piece values, tallying exchanges | Cycles 2 and 3 |
| Counting | Squares, squares of every size, possible moves | Cycle 3, secondary school |
| Powers | Sissa's chessboard, the Shannon number | Secondary school (4e) |
Finding your way: coordinates
The chessboard is a grid of 8 files (from a to h, left to right for White) and 8 ranks (from 1 to 8, starting from White's side). Each square therefore has a unique name, like a point on a graph: you give the file first, then the rank. Locating on a grid is in the primary school maths curriculum, and it is one of the first lessons of the Class'Échecs programme.
Exercises. 1) Which square is the white king on at the start of the game? 2) Which square is at the intersection of file d and rank 5? 3) How many squares are there in file c? 4) How many white squares does the chessboard have?
Answers. 1) e1. 2) d5. 3) 8. 4) 32 (as many as black squares).
Ranks, files, diagonals
The rook moves along ranks and files, the bishop along diagonals, the queen along both. Children thus discover families of lines: horizontals, verticals, and two families of diagonals. The two long diagonals (a1-h8 and a8-h1) have 8 squares; the shortest, in the corners, have just one.
The starting position also offers a lovely line symmetry: the white and black pieces are arranged symmetrically about the line separating ranks 4 and 5, the white queen on a white square (d1), the black queen on a black square (d8).
Exercise. A queen is placed on d4 on an empty chessboard. How many squares does it control?
Answer. 27: 7 squares on file d, 7 on rank 4, 7 on the a1-h8 diagonal (a1, b2, c3, e5, f6, g7, h8) and 6 on the a7-g1 diagonal (a7, b6, c5, e3, f2, g1).
Parity: the colour of the squares
A bishop moves diagonally, and on a diagonal all the squares are the same colour. A bishop that starts on a dark square will therefore never visit a light square: it is an invariant, a property that does not change whatever you do. The knight does the opposite: at every jump, it changes square colour.
An amusing consequence: to get from one dark square to another dark square, a knight always needs an even number of jumps. From a1 to h8 (two dark squares), it needs at least 6. And to get from a1 to b2, the neighbouring square on the diagonal, it needs… 4 jumps, even though it is right next door! This reasoning by square colour is also the key to the famous mutilated chessboard problem, presented on the page the great chessboard puzzles.
Distances: the king and the rule of the square
The king steps one square in any direction, diagonals included. To get from one square to another, it therefore needs as many moves as the larger of the two gaps, in files or in ranks. From a1 to h8, it takes 7 moves (not 14); from b2 to e4, 3 moves. Mathematicians call this way of measuring distance the Chebyshev distance: on the chessboard, the “circles” around the king are squares.
Hence the rule of the square, well known to players: can a king catch an enemy pawn racing to promotion? Draw the square with one side running from the pawn to its promotion square. If the king is inside this square, or can enter it with the move he is about to play, he catches the pawn; otherwise the pawn becomes a queen. Beware: a pawn still on its starting square can advance two squares, so the square must be drawn as if it were already one square further on.
Exercise. In the position above, the a5 pawn needs 3 steps to reach a8. Can the black king stop it a) if it is White to move? b) if it is Black to move?
Answers. a) No: after a6, the pawn's square runs from a6 to c8, and the black king cannot enter it (after Kd5, a7; Kc6, a8=Q, the king is too far away). b) Yes: Kd5 enters the a5-d8 square; then a6 Kc6, a7 Kb7, a8=Q+ Kxa8.
Arithmetic: the value of the pieces
Pieces are traditionally given a value: pawn 1, knight 3, bishop 3, rook 5, queen 9 (the king is priceless). Claude Shannon, incidentally, uses this scale in his 1950 paper on programming chess. These values are used for tallying up.
Exercises. 1) What is the total value of one side's pieces at the start of the game, without the king? 2) White gives up a rook and takes a bishop and a pawn. Has White won or lost in the exchange?
Answers. 1) 8 pawns (8) + 2 knights (6) + 2 bishops (6) + 2 rooks (10) + 1 queen (9) = 39. 2) White gives up 5 and receives 3 + 1 = 4: White has lost 1 point.
Counting: how many squares on a chessboard?
A chessboard has 8 × 8 = 64 squares, 32 white and 32 black. But if you count all the squares you can draw on it, of every size, you find 204: it is the sum 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64. The detail of the calculation is on the page of chessboard puzzles.
Powers: when the chessboard becomes immense
At secondary school, the chessboard becomes a playground for powers and scientific notation. The two classic exercises are the legend of Sissa, which leads to 264 − 1 grains, and the Shannon number, about 10120 possible games. The académies regularly publish worksheets on the legend of chess for the 4e class (age 13–14). Detailed answers are on our pages Sissa's chessboard and the Shannon number.
Which games to go further?
Chess is not the only game that exercises logic. Draughts, Go and many modern board games also call on anticipation and reasoning. Le Palais des Échecs presents a selection: other board games that exercise logic. To build a complete sequence in class around these notions, see setting up an educational chess project.
Frequently asked questions
Which maths notions do you work on with chess?
Locating by coordinates, geometry (ranks, files, diagonals, symmetry), parity (square colour), distances, arithmetic (piece values), counting and, at secondary school, powers.
How many squares are there on a chessboard?
64 squares, arranged in 8 files and 8 ranks, of which 32 are white and 32 black. If you count all the squares of every size, you find 204.
What is the rule of the square in chess?
It is a method for telling whether a king can catch an opposing pawn. Draw the square running from the pawn to its promotion square: if the king is inside this square, or can enter it with the move he is about to play, he catches the pawn.
Why does the bishop never change colour?
Because it moves diagonally and all the squares on a single diagonal are the same colour. A bishop that starts on a dark square stays on dark squares for the whole game.