Problem of the Week #46: Monday November 20th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
This is another extension of the Mini Sudoku problem last month. Imagine instead of a two dimensional grid, we are now filling a 3D grid made up of nxnxn unit cubes with numbers 1 to n such that every row, column, and depth has exactly one of each number. For example, for n=3, a row, column, and depth are shown below:
a). Find one solution for the n=3 case.
b). How many different patterns are there for the solutions to the n=3 case? For example, here is a pattern below, and different numbers assigned to each colour would result in different solutions of the same patter:
c). How many solutions are there for the n=3 case?
d). Prove that it’s impossible for a solution to the n=3 case to have all four cube diagonals (shown below) containing exactly one of each number.
e). What’s the minimum number of unit cubes that need to be filled in to specify one unique solution for a 3D sudoku puzzle in the n=3 case?
f). Find one solution for the n=4 case.
g). Find all patterns for the n=4 case.
h). How many solutions for the n=4 case have all four cube diagonals containing exactly one of each number?
i). What’s the minimum numbers of unit cubes that need to be filled in to specify one unique solution for the 3D sudoku puzzle in the n=4 case?
j). Share your own problem inspired by this one.
k). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.