# How many Hoshi (Star Sudoku) are possible?

A gifted Sudoku booklet's Hoshi are (almost? - I did only spot checks) all identical, except for number permutation and leads.

An excuse for this would be if there were only one legit Hoshi. How many inequivalent (under number permutation and geometric symmetry) filled out Star Sudokus exist?

An example Hoshi can be found here

Fill the grid so that every framed triangle, every horizontal and every tilted row contains the digits 1 to 9, whereby the middle rows expand over the empty hexagon in the middle and the short rows on the edge are completed with the outer corners of the triangles.

• And, pray tell, what is a star sudoku?
– Bass
Sep 16, 2021 at 19:02
• The triangles that stick out are in the exact same lines as the ones they're adjacent to. So those six pairs are always interchangeable (and consequently, each of them must have at least one clue if the puzzle is unique)
– Deusovi
Sep 16, 2021 at 20:57

## 1 Answer

Using the SAT solver from Google's ORTools:

if one fixes all of the cell values in one of the 6 triangles, there are 2,169,216 possible hoshi solutions. As Deusovi points out, any solution can exchange the star points with the cells that they touch to create another solution to the problem. If one considers these solutions as also equivalent, this breaks the solutions into equivalence classes of size 32 each, as there are 2 options for each of the 5 corners...the sixth is fixed to account for numerical permutation. This leaves 67,788 "inequivalent" solutions.

• I'm surprised that the solutions can be counted by brute force (54 variables with 9 values)...less so that the number is rather smallish (albeit >1 :-) @JeremyDover: In the question linked above, I also mentioned the number of same-number filled hourglasses that are not forced. (The 12 touching at the same point as the forced.) Could you scan the solutions for the max and min value? The Hoshi in my booklet always have 10 of these. Sep 17, 2021 at 8:01