1. Problem Solving

Let a1,...,a7 be an arbitrary arrangement of the numbers 1,...,7. Show that (a1-1)(a2-2)...(a7-7) is even.

2. Assume the contrary, i.e. $(a_1 - 1)(a_2 - 2) \cdots (a_7 -7)$ is odd.

Then all of the factors $(a_1 - 1), (a_2 - 2)$ etc. must be odd. Do you see why?

Since $a_1 -1$ is odd, $a_1$ must be even.
Since $a_2 -2$ is odd, $a_2$ must be odd.
....

Continue along these lines and see if you can arrive at an impossible situation, i.e. a contradiction.