### by Howard Williams

#### Published Sunday July 17 2022 (link)

A teacher is preparing her end of term class test. After the test she will arrive at each pupil’s score by giving a fixed number of marks for a correct answer, no marks if a question is not attempted, and deducting a mark for each incorrect answer. The computer program she uses to prepare parents’ reports can only accept tests with the number of possible test scores (including negative scores) equal to 100.

She has worked out all possible combinations of the number of questions asked and marks awarded for a correct answer that satisfy this requirement, and has chosen the one that allows the highest possible score for a pupil.

What is that highest possible score?

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There is a faster method found by Frits (and then analysed by Jim Randell here) which shows that, of the numbers between $$-n$$ and $$m.n$$ inclusive, $$m(m-1)/2$$ are not possible when $$n>=m$$. The missing values are illustrated here for $$n=15$$ and $$m=7$$:

This gives the solution: