Classical Solution
Solving the 2-to-1 version deterministically requires queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires queries.
This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, queries suffice. If we are unlucky, then the first queries could return distinct answers, so queries is also necessary.
If we allow randomness, the problem is easier. By the birthday paradox, if we choose (distinct) queries at random, then with high probability we find a collision in any fixed 2-to-1 function after queries.
Read more about this topic: Collision Problem
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[I too am in Arcadia.]”
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