## Tuesday, 6 January 2009

### More on Al's Problem

There were a few questions about the problem in my last blog... As I stated I read it in a very old math prize competition, so I am as subject to misinterpret them as anyone else...but since it is MY blog... here is how I interpret it...
The first group walks up to a pile with T cannonballs, removes 72, then takes one-ninth of the (T-72) remaining cannonballs. Subsequent groups do the same with the remaining piles and the numbers assigned.

I will leave the question hanging another day ..but for those who need more... here is a second question if you have solved the first, and await furthur challenges... I think it should be possible to change the pattern of values given to the first few people and change the number of divisions that select cannon balls to any value less than the denominator of the fractional part (9 in this case)..

>

I did the simplest case, two divisions, and found it can be done if the first division takes 63 cannonballs plus 1/9 of the remainder. Then the second division will take what is left (which for the number I used for the total works out to two even shares)... This is NOT the smallest possible number of cannonballs, but that gives away too much.

How about three? Well working backwards from the solution for two, I realized that three would work if the first took 54 + 1/9 of the remainder, and the second took 63 + 1/9 of the remainder... the same pattern for larger numbers of detachments could start easily at 27, 36, 45, etc... and it seems as if it would be possible (and actually somewhat simple if I really understand) to generate a pattern for any number of divisions and any fraction we care to use in place of 1/9. Maybe if there is interest, I will address it later