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!

Consider a domino $(x, y)$ such that $x+y = n-k$. Now, consider another domino $(n-x, n-y)$. We have $n-x + n-y = 2n - (x+y) = n+k$. Now, we can map dominoes $(x, y)$ that sums to $n-k$ to dominoes $(n-x, n-y)$. This mapping is one-to-one and onto (incidentally the inverse of this mapping is itself), hence it is a one to one correspondence. Thus, the number of dominoes that sums $n-k$ and $n+k$ must be the same.
