Ponder This Challenge - June 1999 - 1 plus 1/7 of the medals

Ponder This Challenge:

There was a sports competition lasting N days (and N > 1). After the first day they handed out 1 medal and 1/7 of the rest of the medals. After the second day they gave away 2 medals and 1/7 of the rest of the medals, and so forth.

Finally on the Nth day they handed out the remaining N medals.

How many days did the competition last, and how many medals in total were awarded?

FYI: Although this problem may appear simple at first, the solution is actually fairly complex.

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Solution

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  Let the number of medals remaining before day L be written as XL. We have N = XN. From the way that medals are passed out, for L=1,2,...,N–1 we have XL+1 = (6/7)(XL – L). We solve this recurrence to find Then the initial number of medals must be Because this is an integer, we must have that 6N–1 is a divisor of 7N (N–6). Since 6 and 7 are relatively prime, we conclude that 6N–1 is a divisor of N–6. We claim that N 6, because if N > 6 then 6N–1 > N–6 > 0, so that this divisibility is impossible. Checking the integers N=2,3,4,5,6, we find that the only solution is N=6 and X1=36. We started with X1=36. On the first day, 1+5=6 medals were handed out leaving X2=30. On the second day, 2+4=6 medals were handed out leaving X3=24. Similarly X4=18, X5=12, and X6=6, so that on the last (sixth) day, the remaining N=6 medals were awarded.

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