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Last modified on Mon 19 Apr 2021 12.09 EDT Earlier today I set you the following two puzzles. The first is from the UK’s Mathematical Olympiad for Girls: Painting the houses Each of 100 houses in a row are to be painted white or yellow. The residents are quite particular and request that no three neighbouring houses are all the same colour. (a) Explain why no more than 67 houses can be painted yellow. (b) In how many different ways may the houses be painted if exactly 67 are painted yellow? Solution (a) Let us number the houses 1 to 100 from left to right. Divide the street up into a single house (number 1) and 33 blocks of three neighbouring houses: (2, 3, 4), (5, 6, 7), . . ., (95, 96, 97), (98, 99, 100).

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