Enigma 1554: Many times better
6 March 2012
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From New Scientist #2717, 18th July 2009 [link]
My little nephew and I play a simple guessing game using an ordinary pack of cards. In the first round we lay seven cards face down and he has to guess how many are red cards (a number between 0 and 7). Then, likewise, I have to guess how many are black, but I must choose my number so that the two guesses do not add up to 7 (so that at least one of us will be wrong). Then, on turning the cards over, my nephew earns 1 point for each red card plus a bonus of 10 points if his guess was correct and I earn 1 point for each black card plus a bonus of 10 points if my guess was correct. Then in the second round we repeat the exercise with six cards, then in subsequent rounds with five, four, three, two and one card.
In a recent game I recorded cumulative totals throughout and I noticed that after each of six of the seven rounds, my nephew’s total divided by mine was a whole number, and these six whole numbers were all different. Indeed, there were three consecutive rounds after which these three whole numbers were consecutive and decreasing.
Give our cumulative totals after the round(s) when his total divided by mine was an odd number.