#2             64. NOW CUT THAT OUT!                  


Tired after an exhausting game of draughts in front of the fire, Karen and Simon began discussing the properties of the 8 x 8 board.

“I have a puzzle for you,” said Simon. “Suppose the board is cut up into different pieces, each piece made up of whole squares, what is the greatest possible number of pieces?

“Sixty-four, I suppose,” said Karen sleepily.

“No, I mean different pieces. One piece could be a single white square, another a single black square. Then you could have one piece made up of two adjacent squares – and only one, sine all two-square pieces are alike. But then there are four different three-square pieces ... . Dear, you’re not paying attention!”

“Darling ... !”

And so ended another good puzzle session! What is the largest number of different pieces into which the 8 x 8 board can be dissected?
Hints and strategies

Hint

Solution 

Extensions
HINT

List out the possible shapes starting from the smallest. After that, it is probably mostly trial and error.
HINT 2


SOLUTION

Since there are two possible one-square pieces, one two-square, four three-square, and ten four-square, giving a total of 56 squares, only one further (five-square) piece is possible. Thus eighteen is the largest possible number. One solution is illustrated. Of course it must contain several five-square pieces to make up the total of 64.


EXTENSIONS

1. In this problem we have regarded pieces as ‘the same’ if one can be transformed to another by rotation. What happens if pieces are ‘the same’ if one can be obtained from the other by reflection or rotation?

2. There are many problems relating to the chess board. Use your internet search engine to look up ‘chess board puzzles’. Of course, we are thinking of the chess board here, rather than the game!