# Pure algebra…

There are 5 balls with different unknown weights. We know that the weights of all possible pairs are 16,18,19,20,21,22,23,24,26,27. What are the weights of the ball?

Just to verify, is the sum of the weights 53.5. I set three three variable equations, and solve them? ( I MARKED THE BALLS OF A, B, AND C)

• Close, perhaps the addition was not done quite right. – André Nicolas May 20 '14 at 1:49

In the pairs, each number is mentioned $4$ times. Since the sum of the sums of the pairs is $216$, it follows that the sum of our $5$ numbers is $54$.

The sum of the two smallest is $16$, the sum of the two biggest is $27$, so the middle person is $11$.

But then the smallest plus $11$ is $18$, making the smallest $7$. You can take it the rest of the way.

Remark: With basic "figuring" we can find the only $5$ numbers that could possibly work. However, the calculation by no means uses all the information provided. So after we have found the only possible $5$ candidate numbers, we need to check that the $10$ sums are as given: we might have been lied to. They do work.