Three persons
, are playing the following game:
A
-element subset of the set
is randomly chosen, with an equal probability of each choice, where
is a fixed positive integer less than or equal to
. The winner is
or
, respectively, if the sum of the chosen numbers leaves a remainder of
, or
when divided by
.
For what values of
is this game a fair one? (A game is fair if the three outcomes are equally probable.)
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
A
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![\{1, . . . , 1986\}](/media/m/1/3/4/134f2f21154893911dfeb9d97f8d319f.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![1986](/media/m/6/7/3/67346fead933c8cf8d9a83e622e44fb6.png)
![A,B](/media/m/7/1/7/7174f8a9f33236ee137c01b144237389.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![0, 1](/media/m/6/7/6/67619eb0017961c9b7cb6d0abb37829f.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
For what values of
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)