Question

A test had 12 problems, and each problem is with five marks. Full marks are given for a correct answer, two marks given if there is no answer, and no marks are given for an incorrect answer. Some scores between 1 and 60 or impossible to get on this test. What is the sum of these impossible to get scores?

Answer 1

If there are x correct answers and y unanswered, then the score is 5x+2y
So just list all the combinations where x+y=12 to get all the possible scores.

Answer 2

I think there is only one combination which is (5x2)+(2x1)=12. I don't know what to do from there.

Answer 3

Actually, I oversimplified. We need x+y <= 12
So, there are 13*14/2 = 91 combinations in all
x y score
0 0 0
0 1 2
...
0 12 24
1 0 5
...
1 11 27
2 0 10
2 1 12
...
2 10 30
...

Answer 4

This could be done in one line using the APL language, but it gets complicated just doing some sums because there will be several combinations that produce the same score.

I'd write a small program to do it, but hey -- not my job. I don't see an easy number-theoretic way to solve this. Just brute force.