The proof that 0.999... = 1 is elementary, using just the mathematical tools of comparison and addition of (finite) decimal numbers.
About this solution: This solution is correct and demonstrates a level of completeness. The approach is straightforward and uses only the mathematical tools of comparison and addition of decimal numbers.
The Euler's formula states that for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. This can be proven using the Noah's Ark method, which states that if you have a convex polyhedron with V vertices, E edges, and F faces, then V - E + F = 2.
About this solution: The candidate's solution correctly states the Euler's formula and provides a valid proof by the Noah's Ark method. However, the solution could be more complete by providing more explanation and examples. For instance, the candidate could explain how the Noah's Ark method works and provide additional examples to illustrate the proof.
The Pythagorean Theorem states that the length of the diagonal of a square is equal to the square root of 2. This proof shows that the square root of 2 is irrational.
About this solution: The candidate's solution is correct and demonstrates a good understanding of the problem. The candidate has taken a clear and logical approach to solving the problem.