[cptwunderlich]

This is very fundamental not only in maths but in theoretical computer science in general. Even to prove how "hard" a problem is. E.g. look into (Turing/many-one) reductions [1]

Many problems can be reduced to SAT and then you can employ an off-the-shelf SAT/SMT solver to solve it for you. Etc etc.

But in general, being able to reduce problem A to problem B implies that A is not harder than B. I.e., to show that a problem P is undecidable, you can reduce the Halting problem to P.

TSP is NP-hard and can be reduced to SAT, then solved with a SAT solver. But something like determining whether a player has a winning strategy in unrestricted chess (with an nxn board, for arbitrary n) is in EXPTIME and can't be reduced to an "easier" problem.

Edit: Someone ITT also mentioned ILP (integer linear programming). Also a good example. E.g., many optimization programs can be mapped to ILP.

[1] https://en.wikipedia.org/wiki/Turing_reduction