jmc

First, a solution with a Venn diagram: We build this diagram by working from the inside out. First, we put the 12 in the middle for the 12 students who do all three subjects. We then take care of the other 3 overlap regions by subtracting this 12 from each of the totals of students who did a pair of subjects. Finally, we can find out the number of students who did only math by subtracting from 59 (the number of students who did math homework) the numbers in the math-and-other-subjects overlaps. We can then do the same for the other subjects. Adding all the numbers in the diagram, we see that there are 82 students who did some homework, so there are $100-82=\boxed{18}$ who did no homework at all.

Alternatively, we could solve this problem by first adding the number of students for each subject. But students who do two subjects are counted twice, so we subtract the doubles (students doing homework for 2 subjects). Now, we've added the students who did all three subjects three times (adding them in for each subject) and we've subtracted them three times (once for each "double" a student who did all three subjects is in). So, we haven't counted them at all! So, we have to add these "triples" back in! This gives a total of $$59+49+42-20-29-31+12=82$$ students. There are 100 students total, so there are $100-82=\boxed{18}$ students who did no homework.