Solution to Problem 9: "Rule of 72"

A - B / C + D / E + F / G x H - I = 72

I'll start off by assuming G = 1.  That leaves us with X + F x H - I = 72 for some X that's probably also a one-digit number.  So X+F is something less than 18, and (X+F) x H = 72 + X.

I=2: no; 74 = 2*37

I=3: possible 75 = 15*5

I=4: no; 76 = 19*7

I=5: possible; 77 = 11*7

I=6: no; 78 = 39*2

I=7: no, 79 is prime

I=8: possible; 80 = 16*5 (not 10*8, because I is already 8)

I=9: impossible; 81 = 27*3 (not 9*9, because I is already 9)

The most likely seems to be H = 7, I = 5 so if we can solve A - B / C + D / E + F = 11 with 2, 3, 4, 6, 8, 9, we're golden.  If A - B / C were 1, then you'd get the same solution from A - C / B.  Since the solution is apparently unique, let's search first for solutions where A - B / C is not 1.  My first attempt was 8-2/3 = 2.  From here, we can get 11 with 8 - 2 / 3 + 6 / 4 + 9 = 11.  Therefore, the complete solution is

8 - 2 / 3 + 6 / 4 + 9 / 1 x 7 - 5 = 72

Comments: I didn't look at this problem a second time during the test, and spent almost an entire week trying to figure out how to make it work out.  The solution you see here is exactly how I worked in out in my mind.  It took me 10-15 minutes with paper and pencil after a week's worth of musing and that was with a lot of lucky guesses panning out on the first shot.