Does the first player never lose this numbers game?

Question

There is an even number of numbers in a row. Two players cross out numbers one by one from left or right. It is not allowed to cross out a number in the middle. Only left or right. After all numbers are crossed out they find the sum of numbers they crossed. The winner is the one who has greatest sum. How to prove that if first player plays correctly he never loses to second player?

Answer 1

Here is a simple strategy that guarantees a tie for the first player:

Colour the numbers alternately blue and yellow. Let $B$ be the sum of the blue numbers, and $Y$ the sum of the yellow numbers.

If $B \ge Y$: Always pick the blue number. (When it is your turn to play, the two numbers at the end will always be coloured differently.) Then your opponent will have to choose a yellow number. So you end up with $B$ points, and your opponent with $Y$ points.

If $Y > B$: Always pick the yellow number.

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Note that this is not, in general, the best strategy. For instance, if the numbers are $(2, 1, 1, 2, 1, 1)$, then B = Y = $4$, and this strategy will result in a tie. But you can do better by starting with blue number $2$, and then after your opponent plays, switching allegiance to the yellows. This gives you $5$ points to your opponent's $3$.