**
Theorem.
**
Among all positive integers, the integer 1 is the largest.

Proof | |

Step 1. | Take any integer $n \neq 1$ . |

Step 2. | Since $n^2 > n$ , it is not the largest positive integer. |

Step 3. | Therefore, 1 is the largest integer. |

What is the error (if any) in the above proof?

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. - Sherlock Holmes

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The proof "works" by eliminating all possibilities for the largest positive integer other than $n = 1$ , and then claiming that this remaining choice must hence be the largest positive integer.

However, what is has shown is just that "If there is a largest positive integer, then the only possibility for it (according to this proof) is $n = 1$ ". As such, because there isn't a "largest positive integer", hence the result of "it is $n = 1$ " does not follow.

Sherlock's quote applies to the context of his private investigation cases, where he knows that the scenario has existed and is thus looking for a reason to explain its existence. If so, by eliminating all but one possibility, we know that whatever remains must be the truth. It assumes existence through the real-world context that is implicit in his quote. A common scenario that happens is for people to assume that Brilliant problems are well-phrased, and thus there is a correct answer, so they can proceed by elimination.

In truth, this misconception of "If I'm left with 1 answer, then that is correct" occurs especially in geometry problems. The OP takes the algebraic interpretation, sets up the algebraic equation and solves for a certain value. However, it is possible for the answer to result in a physically impossible scenario. E.g. requires negative length which isn't accounted for, makes a certain angle reflex and so the wrong value is used, etc. This is why I often stress "Can this be achieved?" when reviewing a solution for mathematical rigor.

Here is an example where the solution assumes that there is a unique answer, and so merely calculates a necessary condition that gives a unique potential answer. For completeness, it has to show that this potential answer indeed satisfies the given initial conditions.