Solution to Problem 20: "Laser Tag"

First off, we need only calculate the number of blocks that would be passed through by a 9 x 6 x 4 cube and then multiply that number by 7 to get our answer.

With the three sides having no factors common to all of them, you're stuck either deriving the formula or being lucky enough to know it.  For an A x B x C solid, you'll pass through 1 + A + B + C - (A,B) - (A,C) - (B,C), where (M,N) is the greatest common divisor of M and N.  (Subtracting out the gcd's accounts for the times where you cross from one block to another at an edge instead of in the middle of a side, and dividing the whole thing by 7 accounts for the times when you cross at a corner.)

So, for us it's 1 + 9 + 6 + 4 - 3 - 1 - 2 = 14, so the answer to the total problem is 14 x 7 = 98 blocks. 

Comments: I had an inkling of how to solve this one, but I was a nervous that if I were wrong I'd lose 5 points instead of gaining 15.  I asked before I started this if I was 75% sure that I could correctly derive the formula myself, and decided that I was not.   Looking back on it, I'm thinking that taking the chance to make points is worth the risk of being wrong (especially with the less punishing scoring rules).  In my case, if I had answered correctly I would have been in 8'th place (instead of 21'st), and getting it wrong would only have dropped me to 23'rd.  Next time, carpe diem for me!