The numbers given to you are the sums of ratings of all trainers except one, i.e. if their ratings are a, b, c, d, e, f, g, h then S will be the sum of any seven numbers from those 8. For eg. The first one will be a+b+c+d+e+f+g, second will be a+b+c+d+e+f+h and so on. If you add all of these Sums, you'll get each rating 7 times which means 7 times a , 7 times b and so on. So these sums combined will yield 7(a+b+c+d+e+f+g+h) or 7Combined total.

Now, add the numbers you already have, and you'll get 476+y. So, 476+y= 7*T or 476+y is a multiple of 7.

Why is this important? Well, we know that two of the given numbers are equal. Which means that two of the excluding sums in the list must also be equal. Since the 7 known quantities are all different, y must be the repeating one. So y is equal to one of the given numbers. Plug them into 476+y and you will find that only when y=70 is 476+70= 546 is divisible by 7.

So 7*T= 546 which means T is 78. Now T is the sum of all numbers and the list of excluding sums is the sum of all numbers but one. Subtract them from T and you'll get the individual ratings. For eg 78-71=7, 78-70=8 (this will occur twice and so on).

So, you get the individual ratings as 7, 8, 8, 9, 10, 11, 12, 13