Plus or Minus

In the simple game below you start with a sequence of numbers. Taking turns you and your computer place either plus or minus sign between any two successive numbers. When all empty spaces have been filled, the arithmetic expression is evaluated. You win if the result is odd. If the result is even, your computer is the winner.

There is a button "+" at the bottom of the applet. Pressing it will toggle "+" and "-". This is to tell you what sign will be inserted upon your click. If the Random checkbox is checked, the applet selects randomly the length of the sequence. Otherwise, it's up to you: try playing short (even very short) sequences first.

Explanation

References

1. D. Fomin, S. Genkin, I. Itenberg, Mathematical Circles (Russian Experience), AMS, 1996

The parity of the result only depends on the number of odd numbers in the sequence. If the latter is odd the result will be always odd regardless of the distribution of signs. To see this pick up a number A. Add and subtract from it another number B: C₁ = A - B and C₂ = A + B. Then, since C₂ - C₁ = 2B, C₁ and C₂ are always of the same parity, i.e., either both odd or both even. Thus, the selection of signs really does not affect the parity of the result. The latter will be odd iff the sequence contains an odd number of odd members.