MathDB

Let

\mathcal{F}

be a finite family of subsets of some set

X{}

. It is known that for any two elements

x,y\in X

there exists a permutation

\pi

of the set

X

such that

\pi(x)=y

, and for any

A\in\mathcal{F}

\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.

A bear and crocodile play a game. At a move, a player paints one or more elements of the set

X

in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in

\mathcal{F}

in his own color loses. If this does not happen and all the elements of

X

have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.

Problem Statement