There's a common technique in couting/tiling problems in Olympiad which is to use complex numbers. I'm giving some examples:
- (IMO 1995 P6) Let $p > 2$ be a prime number and let $A = \{1, \cdots, 2p \}$. Find the number of subsets of $A$ each havinig $p$ elements and whose sum is divisible by $p$.
The solution to the above problem is to let $\zeta = e^{\frac{2i\pi}{n}}$ and let $x_j$ be the number of subsets of $A$ such that $|X| = p$ and $m(X) \equiv j \mod p$ ($m(X)$ denotes $\sum_{i \in X} x$). Now the rest of the solution is based on using $\sum_{0 \leq j \leq p-1} x_j \zeta^j = [X^p](X+\epsilon)(X+\epsilon^2) \cdots (X+ \zeta^{2p}) = [X^p](x^p+1)^2$ and the lemma that if $a_0 + a_1 \zeta + \cdots + a_{p-1} \zeta^{p-1} = 0$, then $a_0 = a_1 = \cdots = a_{p-1}$.
There's another problem which uses complex numbers, but in a different way
- (Baltic Contest 1998) Can we tile $13 \times 13$ table from which we can remove the central unit square using only $4 \times 1$ and $1 \times 4$ rectangles ?
The answer is no, and the solution is based on putting the number $i^{k+2j}$ on the cell $(k, j)$, and then summing that in two ways to derive a contradiction.
My question is while these applications are cute, they're looking just like a random trick which happens to be useful rather than part of a more general thing.
Is there a way to motivate such applications of complex numbers ? (For example, invariants are also a common trick in olympiad combinatorics problems which involves some process. But some invariants, which seemingly falls from the sky, can often be well motivated using group theory )
Is this "trick" part of a more general technique which is very useful ?