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| Pumpkins 7 (Posted on 2016-02-05) |
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Seven pumpkins are placed in a circle. The seven pairs formed by adjacent pairs of pumpkins are weighed, much like in prior pumpkin puzzles. The weights recorded, sorted in ascending order, are 126, 149, 152, 160, 162, 182, and 191 pounds.
The same seven pumpkins (still in the same order) are weighted in pairs again, this time taking pairs separated by one pumpkin. (If A, B, and C are consecutive pumpkins in the circle then A+C is weighed.) The weights recorded, sorted in ascending order, are 125, 138, 163, 169, 170, 173, and 184 pounds.
From these two sets of weights determine the weights of the seven pumpkins and the order they were placed in the circle.
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Submitted by Brian Smith
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Rating: 4.0000 (1 votes)
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Solution:
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Let the pumpkins be A-G going around the circle. To solve this problem use the identity (A+B) + (C+D) = (A+C) + (B+D). The first set of weighings are potential sums for the left side and the second set for the right side.
There are 21 different ways to form pairs of sums from the first list. Those sums repersent possible combinations for (A+B) + (C+D). The sums are: 275, 278, 286, 288, 301, 308, 309, 311, 312, 314, 317, 322, 331, 334, 340, 342, 343, 344, 351, 353, 373
Similarily, there are 21 different ways to form pairs of sums from the second list. Those sums repersent possible combinations for (A+C) + (B+D). The sums are: 263, 288, 294, 295, 298, 301, 307, 308, 309, 311, 322, 332, 333, 336, 339, 342, 343, 347, 353, 354, 357
These two lists have nine vaules in common:
288 = 126 + 162 = 125 + 163
301 = 149 + 152 = 138 + 163
308 = 126 + 182 = 138 + 170
309 = 149 + 160 = 125 + 184
311 = 149 + 162 = 138 + 173
322 = 160 + 162 = 138 + 184
342 = 106 + 182 = 169 + 173
343 = 152 + 191 = 170 + 173
353 = 162 + 191 = 169 + 184
Looking at just the (A+B) + (C+D) sums, a list of 7 of those sums can be arranged to form a ring such that all seven pairs of adjacent members have one summand in common:
288 = 126 + 162
353 = 162 + 191
343 = 191 + 152
301 = 152 + 149
309 = 149 + 160
342 = 160 + 182
308 = 182 + 126
Then the pairs A+B, C+D, etc can be read off:
A+B = 126
C+D = 162
E+F = 191
G+A = 152
B+C = 149
D+E = 160
F+G = 182
This system simplifies to A=70, B=56, C=93, D=69, E=91, F=100, G=82. The same process could be done with the (A+C) + (B+D) weights to get the same final result.
Other than cyclic rotations and reflections the solution is unique: the pumpkins weights in order around the circle are 70, 56, 93, 69, 91, 100, and 82 pounds. |
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