Mathematics Interview Questions (IV)
16. Sketch .
Solution. From question 1, . is clearly positive within its domain, . Hence, if and only if , i.e., . When , , so is the minimum point. Also,
17. Prove is a multiple of 3.
Solution 1. , so , and , i.e., .
Solution 2. Proof by induction. When , which is a multiple of 3. Suppose for all , is divisible by 3, then is also divisible by 3.
18. How many ways there are of getting from one vertex of a cube to the opposite vertex without going over the same edge twice?
Solution. 30 ways.
The most direct route is via 3 edges: 3 choices for the first edge, 2 for the second, 1 for the third = 6 routes.
The next most direct route is via 5 edges. These routes can be derived by swapping out the first move of a 3-edge route for the 3-edge "long way round" move to the same vertex. So that's 6 routes over 5 edges.
The longest route is via 7 edges. In this case both the first and second move in a 3-edge route are swapped out for 3-edge long-way-round routes. Each starting move can be swapped out in 3 ways. That gives 18 routes over 7 edges.
19. What shape there would be if the cube was cut in half from diagonally opposite vertices?
Solution. A parallelogram.
20. Draw .