Chess Mental Arithmetic

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In the classroom, teachers are always looking for fresh ways to present arithmetic. Chess lends itself to this objective in several ways. For mental arithmetic exercises, use the “piece values” which are conventionally expressed as follows:

Queen = 9   Rook = 5   Bishop = 3   Knight = 3    Pawn =1

The King is missing from this list because either it is worth nothing (one cannot exchange the King for another piece) or because its worth is infinite (without the King there is no game).

Chess pieces can be held up for inspection by the class. “What is a Queen and Rook worth?” “Thirteen” comes the incorrect response.  “Would you swap two Bishops for a Rook?” “No, because two Bishops are worth six points and a Rook is only worth five points.” “Good answer” says the delighted teacher, the children having made the connection between the values of the pieces and decisions to be made during the game. In a real chess game, one would need to take into account other factors, but material balance is always going to be an important consideration.

Let’s introduce subtraction.  Suppose we say the Black pieces are positive and the White pieces are negative. Hold up a Black Queen, two White Knights and two White Pawns. “How many points am I holding?” “Please, I know the answer. It’s one. It’s nine minus six minus two gives one point.”  “Well done!” the teacher purrs.  Hold up a White Pawn. “What am I holding?”  Hands up, “Minus one.”  “Very good, now you know what a negative number looks like.” Looks of gladness dart across the room.

Another topic that pops up is parity – odd and even numbers – which is fruitful when you are least expecting it. “Give me three pieces that add up to the value of a Queen.” That’s easy.  How about three minor pieces (the Bishops and Knights are the minor pieces). Or a Rook, minor piece and a Pawn. It is when children find an answer trivial – they grasp the number bonds – that you move on to the next stage.

Give me four pieces that add up to the value of a Queen.” “That is so easy”, says the unreflective respondent who can add up to nine. “It’s a Rook and four Pawns.” “Sorry, wrong answer – that is five pieces – try again.” The teacher is loving this. An enthusiastic pupil proposes “two Bishops and three Pawns.” before withdrawing having realised that this totals five pieces. “So is there any solution?” asks  the teacher comfortable as one who already knows the answer to a tricky problem. It is a bright child who gets the idea.

What do you notice about the piece values: 9, 5, 3, 1?”  That’s right, they are all odd numbers. When you add two odd numbers you get an even number. When you add four odd numbers you get an even number.  So if you add together the values of four pieces you will always get an even number – you will never get the number 9 which is an odd number.

There is always going to be a smart one who finds an answer anyway. Remember how we said the King could be worth nothing. If we take that to mean the King is worth zero then we can solve the Four Pieces Adding up to a Queen Problem.

About John Foley

Director, London Chess Conference Member, Education Commission, European Chess Union Promoting chess as a way to develop thinking skills

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