Pi Snakes! (Sudoku puzzle)

Back in early 2020, I designed a series of Sudoku puzzles to give myself and my friends a little intellectual stimulation during the lockdown.

In honor of Pi Day I thought I would pull one of them out of the archives.

Heads up: this one is pretty tough! A little math and a lot of logic is involved.

How to play

This puzzle contains pi snakes. The head of a pi snake is a 3; the segments that follow contain the subsequent digits of pi.

(For reference, pi is 3.1415926535897932384626433…)

A snake moves orthogonally, not diagonally. A snake does not touch itself (either orthogonally or diagonally) but may touch another snake orthogonally or diagonally. A snake may not cross or overlap another snake.

Parts of snakes are shown with green-shaded cells. Multiple separate parts may belong to the same snake. An arrow in a shaded cell points toward the tail end of the snake. Not all cells in the grid are part of a snake.

Summing the numbers within a shaded region gives the total length of the snake it belongs to. Here are three examples of what a length-5 pi snake could look like:

Additional rules:

1. Normal Sudoko rules apply.
2. A snake that starts in one 3×3 box may cross into one or more other boxes.
3. However, only one of the 9 boxes may contain parts of two different snakes. The other 8 boxes may contain only one snake (or part of a snake).
4. The bottom left corner contains twice the number of snakes.
5. The two cells in the bottom right corner contain the two-digit total length of all snakes.

Hints:

1. Figure out what the number in the top right green box is. (It can’t be a 1 since that would mean that snake has one digit, and the digit would be 3… which isn’t 1. Could it be a 2? a 3?)
2. Figure out what the number in the upper left green box is. Keep in mind the constraint that that snake can’t share a box with another one (since the middle box already has 2 snakes in it.)
3. You should then know the minimum number of snakes on the board and can figure out the bottom left cell of the puzzle.
4. Figure out where the head of the snake in the central cell could be located, again remembering it can’t go into another box that has another snake in it. That will give you its length and tell you where (some of) the rest of that snake should go.

Enjoy!