# What is the meaning of a Cactus on Day 20 - Zen Garden of the Minesweeper Advent Calendar?

Note: this is not in conjunction with my Minesweeper puzzles

So I was playing Day 20 of the Minesweeper Advent Calendar on heptavegeesimal.com and was reading the instructions when I came across this:

Cactus: Out of the 8 tiles around it the opposite tiles always contain the same amount of mines.

Now, before I did some testing to find out what a Cactus really means, I decided to do some unpacking of what a Cactus really meant:

...the opposite tiles always contain the same amount of mines.

What this means to me (pre-testing): Say we have a Cactus on R4C4. Then it is the same to a green 0 on Day 30 (Diagonal Reverse Synesthesia) (including on orthogonal tiles) because since we have that the opposite tiles always contain the same amount of mines, we must have that there are mines on either

1. R3C4 and R4C4
2. R4C3 and R4C5
3. R3C3 and R5C5
4. R3C5 and R5C3

So, I did some testing, and while it does seem to be similar to the green 0 on Day 30 in the case that both contain 2 mines diagonal to it and that all tiles orthogonal to it are automatically safe, however, it is untrue that opposite tiles orthogonally to a Cactus can have mines. Which means that a Cactus could only have mines at (assuming a cactus on R4C4):

1. R3C3 and R5C5
2. R3C5 and R5C3

So the meaning of a Cactus would be (hopefully a better rewording of the original description):

Cactus: Out of the 4 diagonal tiles, only the opposite tiles can have mines.

with "opposite" in this case meaning you can draw a straight line through the cactus and still hit both mines.

Now, my question is: What is the meaning of a Cactus on Day 20 - Zen Garden, or is my rewording of the description good enough?

Note:

Sorry if my wording confuses you, any feedback on it would be appreciated.

This is actually kind of fascinating, because I think it's a case of selection bias.

Now obviously I don't know how exactly you were testing, but I've spent some time clicking around, dying, restarting, and repeating. Cactuses seem rather rare, but I usually find them once every couple of games (and I also get them confused with the seemingly-much-more-commonly-occurring grass quite a lot). Critically, though, I most often found them on my first move, or on another move that opened up a large area of empty space. We'll come back to why this is important later.

Clicking around the site a bit, I found that day 24 is custom mode, and has a zen garden setting, so I used that to be able to control the variables a little bit more, including making a smaller grid so it would be easier to solve a whole puzzle. And lo and behold, I found a cactus with orthogonally adjacent mines (please ignore the fact that I screwed up the end of the puzzle):

So to answer your actual question, I think the rules for the cactus are exactly as written. To put it another way, if a cell adjacent to a cactus has a mine, the cell opposite that cactus also has a mine, while if a cell adjacent to a cactus is clear, the cell opposite that cactus must also be clear.

Now to come back to the selection bias. As I said, when I was doing my initial testing on the original day 20 board, I most often found cactuses at the corner of a large open region. (For those that have somehow never played minesweeper, if you reveal a cell which does not have a mine in it or anywhere adjacent to it - i.e. one that does not have a number or any other symbol in it - the game will automatically reveal each of the cells around it, which can have a cascading effect and open up a large region of the board.)

To realize why this is important, consider a cactus like the one above, where it has a mine both to its left and right. In order for that cactus to be revealed in a chain effect as described above, it must have an adjacent cell which itself is not adjacent to any mines. However, as you can see here, all of its cells must be adjacent to a mine:

This is not the case, however, if a cactus has mines diagonally adjacent to it:

Therefore, you're much, much more likely to run into a diagonal cactus early in the game than you are an orthogonal cactus.

I can think of three things the not-very-clear wording could mean. Suppose we have a 3x3 block of tiles like this:

a b c
d * D
C B A

Then it could mean (1) A=a, B=b, C=c, D=d (i.e., for each of these pairs either both are mined or both are unmined), or (2) A+a = B+b = C+c = D+d (i.e., all these pairs have the same total number of mines), or much less likely (1') A=a, B=b, C=c, D=d where these values are not the number of mines on a square but the number of mines adjacent to a square, i.e., the number that could be shown on that square if revealed in the game, or (2') A+a = ... with the values again being "number of adjacent mines".

I played a couple of games. Cacti seem to be very rare but I did encounter this configuration where C means a cactus, # means a mine, and . means no mine:

. . . # .
. . . . .
. # C # #
. . . . .
. . . . .

This does obey condition (1): d,D are mined and a,A, b,B, c,C are not. It doesn't obey condition (2) because d+D have two mines and a+A have none. It doesn't obey condition (1') because b has 3 mines next to it and B has only 2. It doesn't obey condition (2') because b+B have 3+2 adjacent mines and a,A have 1+2.

So I think a cactus means: for any tile adjacent to the cactus, that tile has a mine if and only if the corresponding tile on the opposite side of the cactus has a mine.