MathDB

Three persons

A,B,C

, are playing the following game:
A

k

-element subset of the set

\{1, . . . , 1986\}

is randomly chosen, with an equal probability of each choice, where

k

is a fixed positive integer less than or equal to

1986

. The winner is

A,B

C

, respectively, if the sum of the chosen numbers leaves a remainder of

0, 1

, or

2

when divided by

3

.
For what values of

k

is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Problem Statement