Player 1 finds an integer m by taking a permutation {b(j)} of the

first n positive integers,

then forms m:

m = sum{k=1 to n} k * b(k).

Player 2 then, in some time limit, tries to find ANY permutation of

the first n positive integers which also sums to m.

Players take turns making up puzzles and trying to solve other

player's puzzles.

( It is advantageous for the proposer to come up with m's with

relatively few solutions, but are not too easy {easy, such as b(k) = k

or = (n+1-k), which each are the only solutions to their sum m, but

are easy to solve}.)

2 things:

1) One can play a solitaire version of this game, where the

permutation (of 1 through 10, for example) is picked by shuffling

cards, and the player tries to come up with a solution which is

a) a derrangement of the cards' permutation;

b) is not the inverse perm. of the card-perm..

2) How can we determine the number of permutations (for a given n)

which have a given sum m?

thanks,

Leroy

Quet

## Thursday, September 18, 2008

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