Problem of the Week #44: Monday November 6th, 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.
We have previously explored sudoku puzzles in this problem. Today let’s explore one of the deviations from the standard sudoku rules that every row, column, and subgrid (excluded so we can generalize to non-perfect-square side lengths) must have exactly one of each number from 1 to n, where n is the side length of the sudoku puzzle. Let’s include the new rule that a valid solution to a sudoku puzzle must also have one of each number in each of its two diagonals.
For this problem, let’s count all substitution variants that we mentioned in the mini sudoku problem as different solutions (unless otherwise stated in individual questions), but let’s also define these other types of variants:
– Two solutions are rotation variants of each other if one can be rotated to look exactly the same as the other.
– Two solutions are reflection variants of each other if one can be reflected across a vertical, horizontal, or diagonal line to look exactly the same as the other.
For example, in the 2×2 sudoku example from the last sudoku problem, the two solutions to the 2×2 sudoku are rotation and reflection variants of each other besides being substitution variants.
a). How many solutions to the 3×3 sudoku are there using this new diagonal rule?
b). How many solutions to the 4×4 sudoku are there using the diagonal rule if we count all types of variants as the same solution?
c). How many solutions to the 4×4 sudoku are there using this diagonal rule?
d). How many solutions to the 5×5 sudoku are there using the diagonal rule if we count all types of variants as the same solution?
e). Choose one solution from part d) and state how many variants of each type it has.
f). How many solutions to the 5×5 sudoku are there using the diagonal rule?
g). How many solutions to the nxn sudoku, where n is a natural number, are there using the diagonal rule if we count all types of variants as the same solution?
h). For a solution in part g), how many variants of each type does it have?
i). How many solutions to the nxn sudoku, where n is a natural number, are there using the diagonal rule?
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.