Tessellate S.T.E.M.S - Mathematics - School - Set 2 - Problem 2

Let's write the numbers in a $3\times 3$ grid. $\begin{matrix} \phantom{-}1& \phantom{-}2&-3 \\ -4& \phantom{-}0& \phantom{-}4 \\ \phantom{-}3&-2&-1 \end{matrix}$ Then three numbers in the grid sum to $0$ if and only if they make up a row, column, or diagonal of the above magic square. So this game is just tic-tac-toe in disguise. As we all know, there is no winning strategy for either player in tic-tac-toe.

Let A arbitrarily select some number, subsequently, let B select its additive inverse, unless A's selection was 0, in which case B should select one of -3, -1, 1, 3.

Now, A shall select some favourable* number in his second turn, which can always be countered by B, simply because A's third selection, in order to sum his all three of his selections to zero, is unique i.e. B shall counter by selecting that unique number.

This process can be repeated, with B repeatedly countering A's moves until all the numbers are exhausted, resulting in a draw.

A, too, can play the same way, and counter each of B's moves in an identical fashion, once more, resulting in a draw.

Thus, it is clear that there is no way either player can be assured of winning the game.

*Favourable number- a number that is not non-favourable. Non-favourable number- A's second selection, such that no third selection exists within the remaining numbers that sums all three of A's selections to zero.