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If you would like to suggest one, email me. I’m always on the look-out for great puzzles. I set a puzzle here every two weeks on a Monday. The trust organises the BMO, as well as many other school maths competitions. Thanks to the UK Mathematics Trust for today’s puzzle. If he wins another nine of the next ten, he finishes with 21/30, which is 70 per cent. If he wins nine of the next ten, that get’s him to 12/20, which is 60 per cent. If he wins one of the next five, that get’s him to 3/10, which is 30 per cent. That get’s him to 2/5, when he will have won 40 per cent. Let’s say that after winning 1/2, Arun wins one of the next three. Let’s now rewrite the other percentages required as fractions: 30, 40, 60 and 70 per cent are 3/10, 2/5, 3/5 and 7/10. So, after two games his running total of wins/games is 1/2, which is 50 per cent. I told you that Arun wins one of the first two games. (i) because there are not enough games to go from 3 to 14 wins in 10 games, and (ii) because you can’t go from 7 wins to 6. Then either (i) after 10 games he has won 3 of them, and after 20 games he has won 14 of them, or (ii) after 10 games he has won 7 of them and after 20 games he has won 6 of them.
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Let’s say the total number of games is 20. We can eliminate the chance that the total number of games is 10, since if it was then this would mean that after ten games Arun would be finishing having won both 70 and 30 per cent of games, which is impossible. In other words there is a w such that 10 w = 7G. However we also know that there is a point at which he has won 70 per cent of games. We can rearrange this equation as 10W = 3G, from which we can deduce that G must be a multiple of 10.
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If at any stage Arun has won 30 per cent of the games he played, then we can write the equation: First, let G be the number of games played so far. What is the smallest possible number of games they played? The results of these calculations were exactly 30 per cent, exactly 40 per cent, exactly 50 per cent, exactly 60 per cent and exactly 70 per cent, but not necessarily in that order. At five points during the day, Arun calculated the percentage of the games played so far that he had won. Arun and Disha played several games of table tennis.