i didn't check if 57 and 7 would work if you round down any fractional number... but i'd say that if the original problem didn't say you can round up or down, you probably would be better off sticking with natural numbers. especially since there is a solution that works! but 57 & 7 would be a cool extension to the problem.
with regards to your equation, i don't think the problem implied that the same number of medals were given out each day, so i don't think you can quite make that assumption.
here's why the last day has to be a multiple of 6 because of the 1/7 condition--it's really obvious once you see it. since on the previous day 1/7 of the the rest of the medals were given out, that means there are 6/7 of medals left for the next day. now, on the last day, you give away all the remaining medals, which has to be 6/7 of some integer. now you see why that last day has to be a multiple of 6!
once you realize that the number of days n = 6*i where i is some integer, of course you proceed to test if i=1 works. turns out it does, and gives a very nice, symmetric solution (6 medals a day for 6 days), so you can bet your money that it's the right answer. however, you still need to show that it's the only solution. this part isn't that hard although it's a little tricky, but essentially you show that only when i=1 you can ensure you get integer value of medals awarded on every single day.
the Wii is an epidemic.
