When asked to show a math problem has a unique solution, students sometimes think that if an algorithm leading to a solution has unambiguous instructions at each step (no need to make choices at any point) then the solution they find has to be the only solution. Loosely speaking, if each step in reaching the solution is "unique" then the end result has to be unique (the only possible solution).
This is not true, and I think it would be nice to have a list of examples at different levels (of undergraduate mathematics) showing why this idea is mistaken.
For example, if asked to solve $55x+32y = 1$ in integers then Euclid's algorithm for computing $\gcd(55,32)$ followed by back-substitution (reversing the steps of Euclid's algorithm) is a procedure where each step is completely determined by the previous ones and leads to the definite answer $(x,y) = (7,-12)$, but the original equation has infinitely many integral solutions: $(x,y) = (7+32t,-12-55t)$ for integers $t$.
What other examples can people offer? I am not interested only in computational problems.