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Mathematics / Theorems
DifficultyHard
This theorem states that if a and b are positive integers, then their sum a + b is also a positive integer. This is true because both a and b are positive, so their sum must be positive as well.
About this solution: The candidate's solution is correct and demonstrates a level of completeness. The candidate has correctly identified that this is a direct result of the definition of addition for integers. The candidate has also correctly identified that the sum of two integers will always be an integer. Therefore, the candidate's solution solves the problem and is a correct approach.
The given statement is true. If n^2 is even, then n is even.
About this solution: This is a correct solution.
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 Euler 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 seen by looking at a tetrahedron, which has four vertices, four faces, and six edges.
About this solution: This solution correctly states the Euler formula and provides a brief explanation of why it is true. However, it could be more complete by providing a more detailed explanation of the proof. For example, the solution could explain how the fact that every edge is shared by two faces implies that the number of edges is equal to twice the number of faces. Additionally, the solution could provide a specific example to illustrate the formula.
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.
This puzzle asks you to write a function that takes in a list of integers and outputs the largest number in the list.
About this solution: The candidate's solution correctly finds the largest number in the list. The candidate uses a for loop to iterate through the list and compare each number to the current largest number. If the number is larger, it becomes the new largest number. This is a good approach as it iterates through the entire list only once.
Given a list of numbers, find the two numbers with the greatest product and return that product.
About this solution: The candidate's solution is correct and demonstrates a level of completeness. The candidate has correctly identified that the last two numbers in the list will be the largest two numbers and has sorted the list in ascending order. This is the optimal solution.