What are the weights of the five boxes?

Question

A shipping manager wants to determine the weight of five boxes. Each weighs a different integer amount less than 100 kg. Unfortunately, the only scales available measure weights greater than 100 kg. The manager therefore decides to weigh the boxes in pairs so that each box is weighed with every other box. The weights of ten pairs are (in kilograms) 110,112,113,114,115,116,117,118,120, and 121. From this info, the manager can determine the weight of each box. What are the weights of each of the five boxes? .

Answer 1

Important information:

1. "five boxes" -- name them A.B.C.D.E from lightest to heaviest
2. "Each weighs a different integer amount"
3. "weigh the boxes in pairs"; "each box is weighed with every other box. "

The ten pairs are:

A B

A C

A D

A E

B C

B D

B E

C D

C E

D E

The of pairs are given in sorted order: 110, 112, 113, 114, 115, 116, 117, 118, 120, 121

Looking at the list of weights, we observe that there must be both sequential and non-sequential integers in the sorted solution list. So, we don't know which integers ore omitted, but we sorted ABCDE to be lowest to greatest, so we know:

A + B = 110

D + E = 121

[but we don't know the other pairs, yet]

Also, 4A+4B+4C=4D+4E =1156

so, A+B+C+D+E = 289

(110) + C + (121) = 289

C = 58

A+C = 112 [the second smallest total\

A = 54

C + E = 120 [the second largest total]

E = 62

A + B = 110

54 + B = 110

B = 56

54 + 56 + 58 + D + 62 = 289

D = 59

To check:

Answer 2

name the boxes A,B,C,D,E

Each unique pair can be arbitrarily assigned to one of the given values

A+B = 110

A+C = 112

A+D = 113

A+E = 114

B+C = 115

B+D = 116

B+E = 117

C+D = 118

C+E = 120

D+E = 121

If we start combining things, a problem emerges---for example:

(A+E) - (A+D) = E-D = 1

But:

(C+E) - (C+D) = E-D = 2

This suggests that the given sums are not correct. My hunch is that the sums were supposed to be sequential in steps of 1. Let's see if someone else has a better idea....

Answer 3

Suppose that the pairs were in this order, but we could have assigned any of the pair weights to any of the pairs for the purpose of this example:

From #10, 4 and 5 together weigh 121 and from #9, 4 and 5 together weigh 120, so from 9 and 10, we can see that 3, 4 and 2 5's weigh 241, but from #7, we see that 3-4 weigh 118 so 5 weighs (241-118)/2 = 61.5. If 5 is 61.5, from 10, 4 is 59.5, from #8, 3 weighs 58.5. From # 5, 2 weighs 56.5, and finally, working up the list, from #1, 1 weighs 53.5

So yes, it is possible, and the weights are in bold above. Check for yourself using another pattern!

Answer 4

Since the order of the weighings was not specified, I shall create one.

The five boxes are A, B, C, D, and E

A + B = 110

A + C = 112, C = B + 2

A + D = 113, D = C + 1

A + E = 114, E = D + 1

B + C = 115

B + B + 2 + 115

2B = 113

B = 56.5

From here can you calculate the other four weights?