Problem of the Week 1090

A Painted Cube

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Solution

The problem leads to finding to roots for the equation ...

6(n - 2)² = (n - 2)³

... and there are two distinct roots: 2 and 8. A proper solution *must* address both of these roots and either accept the trivial n = 2 as a reasonable solution or reject it. In my opinion, you can make a reasonable case either way about n = 2. I leave it to you to decide where you fall.

In short, a solution that directly addresses the n = 2 case is more complete.

In response to the solution, some people suggested that if n = 2 is legitimate then n = 1 (there are no cuts) and n = 0 (there is no cube) should be solutions, as well. If you decide that you accept the trivial n = 2 solution, then you also get to decide whether you like these solutions. I prefer to interpret the problem as (1) the cube does exist and (2) Alice did cut it.

[Back to Problem 1090]

© Copyright 2008 Stan Wagon. Reproduced with permission.