Enigma 1097: Chessboard triangles
18 September 2017
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From New Scientist #2253, 26th August 2000 [link]
Take a square sheet of paper of side 1 kilometre and divide it into small squares of side 1 centimetre. Colour the small squares so as to give a chessboard pattern of black and white squares.
When we refer to a triangle, we mean a triangle OAB, where O is the bottom left corner of the square of paper, A is on the bottom edge of the paper and B is on the left hand edge of the paper.
Whenever we draw a triangle then we can measure how much of its area is black and how much is white. The score of our triangle is the difference between the black and white areas, in square centimetres. For example if OA = 3 cm and OB = 2 cm then we find the score of the triangle is 1/6 cm².
Question 1. What is the score of the triangle with OA = 87,654 cm and OB = 45,678 cm?
Question 2. What is the score of the triangle with OA = 97,531 cm and OB = 13,579 cm?
Question 3. Is it possible to draw a triangle on the paper with a score greater than 16,666 cm²?