# Another numbers game (reversed)

Discrete Mathematics Level pending

The numbers $$-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7,8,9,10$$ are written on a blackboard.

On each turn, Anna picks two numbers, and Bob erases these two numbers, and then adds their product, sum or positive difference to a separate total.

For example, If Anna picks 2 and 3, Bob erases them and decides to multiply them, so the separate total is $$2\times 3=6$$. If Anna picks 1 and -1 next, Bob erases them, decides to take the positive difference ($$1-(-1)=2$$) and adds it to the total ($$6+2=8$$).

Eventually, there will be no numbers left on the board. This time, Bob is trying to minimise the total and Anna is trying to maximise it.

If Anna and Bob play optimally, what will the total be?

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