In the expression $1 * 2 * 3 * . . * 9$, the asterisks are replaced with minus or plus.

a) Can the result be 0?

b) Can the result be 1?

c) What numbers can result?
a) An even number cannot be obtained, since the number of odd addends (which is 5) is odd.

c) To get the number 45, all asterisks must be replaced with plus signs, and to get the number -45, with minus signs.

We will prove that if we can obtain a number $n<45$, then we can change the placement of the signs so that we get $n+2$.

If a minus sign precedes the one, changing it to a plus will increase the sum by 2. If a plus sign precedes the one, then somewhere in the expression, a plus is followed by a minus (otherwise, all signs would be pluses, and the sum would be 45). Changing both of these signs will again increase the number by 2 (instead of

$+k-(k+1)$, we get $-k+(k+1))$.

## Answer

a) It cannot; b) it can; c) all odd numbers from -45 to 45.

## Problem