Question 6
Let $m,n,p$ be nonnegative integers, not all zero. Coordinate space is divided into unit cubes by equally spaced parallel coordinate planes. A filling of every unit cube with one of the numbers $1,2,\ldots,60$ is called a Dien Bien filling if the following holds: in every rectangular box formed by the grid planes whose three side lengths, in unit cubes and in any order, are $2m+1,2n+1,2p+1$, the number in its central unit cube is the arithmetic mean of the numbers in its eight corner unit cubes. How many Dien Bien fillings are there?
The source adds: two fillings are the same if, after choosing a cube as the center in each, cubes having the same position relative to the chosen centers carry equal numbers. The distinction between a fixed origin and equivalence under translation is discussed explicitly in the solution.
The source adds: two fillings are the same if, after choosing a cube as the center in each, cubes having the same position relative to the chosen centers carry equal numbers. The distinction between a fixed origin and equivalence under translation is discussed explicitly in the solution.