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 Contestants in a Circle (Posted on 2011-01-31)
The eight contestants were seated at a circular table in alphabetic order clockwise: Alice, Bob, Carol, Dave, Eve, Frank, Gisele and Harry. Some points had already been awarded by the judges, in varying amounts to the different competitors, but the competitors didn't know what their totals were. As a last competition, they were to deduce all their scores thus far based upon the announcement of eight totals of three contestants' scores, each total being the sum of three successive people around the table, so for example, the total for Harry, Alice and Bob (the triplet centered on Alice) was given as 38.

The successive triplets centered on Bob through Harry totaled as follows: 49, 45, 48, 52, 44, 39 and 48.

What were in fact the individual scores to that point in the competition?

 Submitted by Charlie No Rating Solution: (Hide) The total for Alice, Bob and Carol, 49, is 11 higher than that the total of 38 for Harry, Alice and Bob, so Carol's score thus far was 11 higher than Harry's. Going around the circle, we find similarly, using their initials as their totals: ```D - A = -4 E - B = 3 F - C = 4 G - D = -8 H - E = -5 A - F = 9 B - G = -10 ``` Relative to Harry, then, the other scores are: ```A = H + 24 B = H + 2 C = H + 11 D = H + 20 E = H + 5 F = H + 15 G = H + 12 ``` The first of the above was of course C = H + 11, already mentioned; then, as F = C + 4, it follows that F = H + 15; etc. The total of all the numbers given, 38, 49, 45, 48, 52, 44, 39 and 48, was 363, which was three times the total of the people's scores, which was therefore 121. But, counting Harry, the total of the scores is 8*H + 89, so Harry's score, H, is 4, and the other scores can be obtained from the table above: ```H = 4 A = 28 B = 6 C = 15 D = 24 E = 9 F = 19 G = 16 ``` Based on Enigma No. 1620, "Pieces of eight", by Gwyn Owen, New Scientist, 6 November 2010, page 28.

 Subject Author Date re: easy way out broll 2011-02-01 02:39:30 easy way out Ady TZIDON 2011-01-31 18:21:11 Possible solution broll 2011-01-31 15:20:03

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