, 2018

For the product of two integers to equal 1, the two integers must both equal 1 or must both equal $-1$.

Similarly, if the product of six integers is equal to 1, then each of the six integers must equal 1 or $-1$.

For the product of six integers, each of which is equal to 1 or $-1$, to equal 1, the number of $-1$s must be even, because an odd number of $-1$s would give a product that is negative.

That is, there must be zero, two, four or six $-1$s among the six integers.

We summarize these four possibilities in the table below.

Number of $-1$s
Product of the six integers
Sum of the six integers

0
$(1)(1)(1)(1)(1)(1)=1$
$1+1+1+1+1+1=6$

2
$(-1)(-1)(1)(1)(1)(1)=1$
$(-1)+(-1)+1+1+1+1=2$

4
$(-1)(-1)(-1)(-1)(1)(1)=1$
$(-1)+(-1)+(-1)+(-1)+1+1=-2$

6
$(-1)(-1)(-1)(-1)(-1)(-1)=1$
$(-1)+(-1)+(-1)+(-1)+(-1)+(-1)=-6$

Of the answers given, the sum of such a group of six integers cannot equal 0.