Five pumpkins are weighed two at a time in all ten sets of two. The weights are recorded as 16, 18,19, 20, 21, 22, 23, 24, 26, and 27 pounds. All individual weights are also integers.
How much does each pumpkin weigh?
I solved this one in just a couple of minutes by reasoning:
All pumpkins are different weights, since we have 5 choose 2 = 10 different results for the weighings. The two lightest one total 16. The two heaviest one total 27. The weight of the middle pumpkin can be determined by adding the total of all the weighings = 216. Divide by four since each individual is weighed four times = 54. Deduct the two heaviest =27, and the two lightest =16, and the weight of the middle remains = 11. Then to not produce duplicate weihings we need the differencew between the two lightest and two heaviest to be different, so try 9 and 7 for the lightest ones (difference of 2) and 12 and 15 for the heaviest ones (difference of 3). My solution is: 7,9,11,12,15