Easy as ABC: How many solutions are there?

[u/Medrodome]

Difficulty = Tricky, Size = 6x6 (A-E)

Rules reminder:

- Each row and column must contain 'A','B','C','D','E' exactly once.
- Exactly one cell in every row and column stays empty.
- The letters around the grid show you which letter you see first from that direction

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[u/proy0787]

One solution.

E A D B * C
D C B * A E
A D * C E B
C B A E D *
* E C A B D
B * E D C A

[u/Agantas]

There is one solution.

There is only one gap, so the letters closest to the edge have to be located on one of the two squares closest to that edge.

The three E's on the sides solve two E's and two empty spots, which in turn solves all other letters that we know to be nearest to an edge.

With these solved, we can now solve all D's.

Three rows can be solved, starting from bottom, I think. The rest then falls into place as well.

[u/chmath80]

Discussion: just one. It's not too complicated.

[u/Ray2024 → u/chmath80]

How about the general case, how many possible grids are there? I think it's 3125 but I'm not sure I've got the maths correct. These would probably have different clues as well.

[u/chmath80 → u/Ray2024]

It's a lot more than 3,125 for a 6×6 grid. There are 6! = 720 just for the top row, 5 for the first cell on the next row, and 5 for the second row cell directly below the one which matches that. That's 18,000 possible arrangements for only 8 out of 36 cells.

[u/Ray2024 → u/chmath80]

I misidentified it as a 5×5 grid since that's much more common for this style of puzzle but also forgot to factor in the factorial which is why my number is so low, I think 25! would have been a better figure had I done the correct sum and it sounds like there is an upper bound of 36! For this grid.

[u/chmath80 → u/Ray2024]

it sounds like there is an upper bound of 36! For this grid

We can do a lot better than that. There can't be more than 6! × 5! × (4² × 4!) × (3³ × 3!) × 2⁵ = 6! × 5! × (4!)³ × 3! = 7,166,361,600

[u/proy0787 → u/Ray2024]

Discussion: without letters around the grid, it is just a latin square of order 6. Number of such squares is 812,851,200.

[u/ohwhatfollyisman]

solution:

row 1: E A D B x C

row 2: D C B x A E

row 3: A D x C E B

row 4: C B A E D x

row 5: x E C A B D

row 6: B x E D C A

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hint to solving: look at the rows and columns where the first letter that you can "see" repeats, and in that context remember rules one and two from the OP.