A shipping clerk wishes to determine the weights of each of the 4 boxes. Each box weighs a different integer amount less than 70 kg. Unfortunately, the only scales available measure weights in excess of 70 kg. The clerk therefore decides to weigh the boxes in pairs so that each box is weighed with every other box. The weights for the 6 pairs of boxes are (in kg) 84, 85, 87, 88, 90, and 91. From this information, the clerk can determine the weight of each box.

Question

A shipping clerk wishes to determine the weights of each of the 4 boxes.  Each box weighs a different integer amount less than 70 kg.  Unfortunately, the only scales available measure weights in excess of 70 kg.  The clerk therefore decides to weigh the boxes in pairs so that each box is weighed with every other box.  The weights for the 6 pairs of boxes are (in kg) 84, 85, 87, 88, 90, and 91.  From this information, the clerk can determine the weight of each box.

what are the weights of the four boxes?

MathMate · Answer

I do not know if there are multiple solution, here's what I've found.
The sums of weighed pairs are all correct, but the individual weights are not integers.

A=81/2
B=87/2
C=89/2
D=93/2

A+B=84
A+C=85
A+D=87
B+C=88
B+D=90
C+D=91