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My solution is below. Bractals presents a solution here.
Let A>B>C>D>E and let w represent the repeated weight. The following equations can be formulated:
A+B=88 - sum of the two heaviest weights
A+C=84 - sum of the heaviest and middle weights
C+E=56 - sum of the lightest and middle weights
D+E=52 - sum of the two lightest
4*(A+B+C+D+E) - w = 636 - the sum of all the weighings
w can be any of 60, 68, 72, 76 or 80. Substituting those values for w yields five different sets for A,B,C,D, and E. Each set of values is summed in pairs to see if that set gives the same nine values as the original values.
w | A | B | C | D | E |
---+---+---+---+---+---+-----
60| 50| 38| 34| 30| 22| No pair sums to 76
68| 48| 40| 36| 32| 20| Solution
72| 47| 41| 37| 33| 19| No pair sums to 68
76| 46| 42| 38| 34| 18| No pair sums to 68
80| 45| 43| 39| 35| 17| No pair sums to 68
Since 48, 40, 36, 32, 20 is the only set of values which sums properly, it must be the solution. |
