How to prove the amounts of dominos with x+y=n+k = x+y=n-k? I've been trying to answer this question for hours with no luck at all.. The question is the following:
Question

*

*Imagine we have domino blocks of the following shape: [ x | y ] with x, y ∈ [0..n].

*Imagine 0 ≤ k ≤ n.

Show that the amount of blocks with

*

*x + y = n - k

is equal to the amount of blocks with

*

*x + y = n + k

The question also says:
In both cases, you will find (n - k + 1)/2, but you need to prove this.
Thanks a lot for the help!

For each $x$ from $0$ to $n-k$ let $y=n-k-x$. This makes $n-k+1$ blocks with $x+y=n-k$. But if $n-k+1$ is even, we must count only a half of them, because the block $[x|y]$ is the same as $[y|x]$. And if $x-k+1$ is odd, there is a block that has no pair, namely $[(x-k)/2, (x-k)/2]$.
For each block $[x|y]$ with $x+y=n-k$ we have exactly one block $[n-x|n-y]$, and $n-x+n-y=2n-(n-k)=n+k$, so the number of blocks with $x+y=n+k$ is the same as before.
