## How Much did the Samples Weigh?

How Much did the Samples Weigh?

A chemistry professor asked his graduate student to find the weights of five different samples of an unknown material.

The student weighed each possible pair of samples, and recorded the weight of each pair as 120, 110, 117, 112, 115, 113, 114, 121, 116, and 118 grams.  When he tried to figure out the five individual weights, he realized that he had neglected to record which samples made up each pair.

The professor said that he could figure out the five weights from the data the student had already collected.

How much did the samples weigh?

First, no two of the samples have the same weight. Proof: Suppose x1 = x2.  Then x1 + x3 = x2 + x3, and two of the pairwise weighings would be the same.  Since no such duplication occurs, all samples are different.

Let x1, x2, x3, x4 and x5 be the five samples, arranged from heaviest to lightest.

x1 + x2 = 121, and
x4 + x5 = 110

We know this because every pair other than (x1, x2) contains, at heaviest, one sample equal to x1 or x2, and one sample lighter than either of x1 or x2, so the heaviest pairwise weighing must be (x1, x2). Similarly with (x4, x5).

The sum of all of the pairwise weighings, 1156, is equal to four times the sum of all the samples. This is because each sample participates in four weighings. I'm not going to write out this long equation! So

x1 + x2 + x3 + x4 + x5 = 1156 / 4 = 289

Subtracting the previous two equations from this one,

x3 = 58

Now we're getting somewhere! The second largest pairwise weighing, 120, must be (x1, x3). Every other pair, contains either a sample less than x1, a sample less than x3, or both. A similar argument can be made that (x3, x5) weighs 112.

x1 + x3 = 120, so x1 = 62
x1 + x2 = 121, so x2 = 59

x3 + x5 = 112, so x5 = 54
x4 + x5 = 110, so x4 = 56

and the final group is (62, 59, 58, 56, 54).