Problem of the Week #42: Monday October 23rd, 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.
Sudoku is a one-player game where the player tries to fill the spaces of an nxn grid using numbers 1 to n, where n is a natural number, such that every column and every row contains exactly one of each number. For special cases where n is the square of another natural number k, there is an additional rule that divides the grid into smaller k kxk squares where each contains one of each number. Let’s ignore this additional rule so that we can apply this game to square grids of all sizes.
For example, here are all possible solutions to the 2×2 sudoku:
Notice that if we substitute all the 1s with 2s and vice versa in the first solution, we get exactly the second solution. For simplicity’s sake, if one solution can be made to look exactly the same as another solution through substitution, they are considered the same solution. Let’s call the two representations of it substitution variants of each other. Therefore the 2×2 sudoku has only one solution, and the above are two substitution variants of that solution.
a). Find one solution for the 3×3 sudoku.
b). Find a substitution variant of the solution you found in part a).
c). Find a different solution for the 3×3 sudoku.
d). How many different solutions are there for the 3×3 sudoku?
e). What’s the minimum number of spaces that need to be filled in before we can determine it to be one specific solution for the 3×3 sudoku?
f). What’s the minimum number of spaces that need to be filled before we can determine it to be one specific substitution variant for the 3×3 sudoku?
g). How many different solutions are there for the 4×4 sudoku?
h). How many substitution variants does a solution have in the 4×4 sudoku?
i). What’s the minimum number of spaces that need to be filled in before we can determine it to be one specific solution for the 4×4 sudoku?
j). What’s the minimum number of spaces that need to be filled in before we can determine it to be one specific substitution variant for the 4×4 sudoku?
k). Share your own problem inspired by this one.
l). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.