# Can you create a Minesweeper puzzle with 'information crossover'?

The object of this puzzle is to construct a Minesweeper grid with a special property. The grid can be any shape, but it must have four labeled squares around the outer edge. Like this:

The four labeled squares must be labeled A, B, C, D, in that order going around the outer edge.

The objective is for a Minesweeper player to be able to conclude that A and C are the same, and B and D are the same, but not be able to conclude anything else. To be specific:

Place numbers in the grid to get a Minesweeper puzzle. This puzzle must have the following properties:

1. There are no safe clicks. That is, for any unnumbered square, there is a solution in which that square is a mine.
2. In any valid solution, A=C and B=D. (A=C means that A has a mine if and only if C does.)
3. A valid solution must satisfy every combination of A=C and B=D simultaneously.

Again, the grid can be any shape. The grid can even have holes in it, but understand that "outer edge" does not include the edge neighboring a hole. A, B, C, D can be placed anywhere along the outer edge, but must be in that order. Aim for the smallest grid!

• When you say A=C .. do they count each other for determing number of mines next to ? So if there are 2 mines next to A. and 1 next to C .. does A show 2, or 3 ? does C show 1 or 3 ? Commented Mar 24, 2015 at 15:59
• @Ditto What it means is that either A and C both have mines, or neither one does. What number they would show if they didn't have mines is not part of the puzzle. Commented Mar 24, 2015 at 16:01
• Ok, so they are different squares .. they just share a mine/no mine property .. got it :) Commented Mar 24, 2015 at 16:01
• You're asking this to get some help completing your P=NP Minesweeper proof, aren't you? ;-) Commented Mar 24, 2015 at 20:10
• Does the definition of the puzzle grid include the number of mines to look for (as in the minesweeper game), or can different solutions use different numbers of mines? Commented Mar 25, 2015 at 1:45

I think I got it now. Using a minesweeper developer program found on here, you can create, evaluate and try your own board. I have uploaded an image of my solution, or else this would've been a LOT of typing:

And an explanation about what you see in the image:

I've put the 4 labels (A, B, C and D) beside each questionmark where it should be. The red flags mark the spots of pre-discovered bombs.
Every pair (A-C and B-D) have their own 'wire' as it is called. Then an intersection to let these signals cross eachother, and arive at the other side.
For the first rule: there are no safe clicks, in all possible solutions this setup has got, any unnumbered square has been a mine at least once.
In èvery solution of this setup: it covers the A=C and B=D requirement. So both the second and third rule are covered too.

1) .A   2)  A  3) AB
1 .    D2B    DC
D1.1B     C
. 1
C.

• Maybe I made the rules unclear. Rule 2 says "In any valid solution, A=C and B=D." Your first option breaks this because there is a solution where A and D have mines and B and C don't. Your second option breaks this because anything is a solution. Is it clear how Rule 2 works now? Commented Mar 24, 2015 at 15:04
• @Lopsy Ok, I think it is clear now, let me (and others) work on a possible solution Commented Mar 24, 2015 at 15:10
• I don't think your new answer implies that A=C and B=D. For example, you could have A with a mine and the bottom-right dot has a mine, but C does not. Commented Mar 24, 2015 at 15:58
• What I'm saying is that there is a valid solution where A is a mine and the dot next to C is a mine, so C is not a mine. So A$\neq$C. Commented Mar 24, 2015 at 16:10
• Btw can we adapt this to make ABCD on the outer edge of the board to fulfill all the requirements? Commented Mar 30, 2015 at 21:26

I think you might need to clarify what a "hole" is.

  A
D   B
C

This satisfies the puzzle's requirements.

There are no safe spaces; all of the spaces are mines. Because none of the spaces are adjacent to each other, none of them can be a "numbered square"; the value of this number would be 0, making it an empty (or safe) space.

• This could be a solution indeed, but there might be a possibility this is not allowed because none of the sides are touching, thus not a minesweeper puzzle. I'll leave it up to Lopsy Commented Mar 24, 2015 at 15:27
• It follows the rules laid out in the challenge and it follows the rules of Minesweeper as they are modified to include holes. It's really just a question of what makes a "smaller grid". 4 non-hole squares is smaller than 8 non-hole squares, but they take up more space if you include the holes in the calculation. Commented Mar 24, 2015 at 15:31
• How does this not have the "anything is a solution" problem that @JBSregath's second answer does? Commented Mar 24, 2015 at 15:33
• @KSmarts: See the second spoiler block. Commented Mar 24, 2015 at 15:34
• The intention of rule 3 is that there should be a solution where (for example) A and C are both mines and B and D are both safe. That is, all 4 mine-settings of A, B, C, D satisfying A=C and B=D extend to a solution. I see now that I phrased this rule badly, sorry to the solvers so far. Commented Mar 24, 2015 at 15:42

I'm probably twisting the rules a bit, but here goes. "The grid can be any shape". Well, my grid is a torus! (And it kinda looks like a swastika, but let's ignore that.)

C+A D
1 +
D1+1B
+ 1
B C+A


The labels are shown twice to indicate where the grid "wraps around". The grid itself uses 11 squares.

I realize that this probably isn't what is intended, but I thought of this and decided to post it while working on a "real" answer.

1A1
D B
1C1

The space in the middle is a "hole". If A has a mine, then B and D do not, which means that C does. Similarly, if A does not have a mine, then B and D do, so C does not. Thus, this satisfies all three conditions.

It uses a 3x3 grid, with a total of 8 squares.

• @Ian Macdonald and KSmarts: I think Lopsy means that for évery combination a solution has to exist. Both AC and BD have mines, AC have mines and BD are clear, AC are clear and BD have mines, both AC and BD are clear Commented Mar 24, 2015 at 15:23
• @JBSregath: no, not quite. The reason why he claimed your answer did not follow rule 2 was that there existed a valid combination of ABCD that did not satisfy rule 2. No such rule-breaking combination exists in this answer (or mine). Commented Mar 24, 2015 at 15:28
• @Ian Macdonald: And rule #3: For any mine-setting of A, B, C, and D that satisfies A=C and B=D, this setting extends to a solution of the whole grid. This means a valid solution has to exist for each of these settings: both AC and BD mines, AC mines and BD clear, AC clear and BD mines, both AC and BD clear Commented Mar 24, 2015 at 15:32
• @JBSregath I can see how it might mean that. I'll wait to hear from Lopsy, though, because that makes it a lot harder. Commented Mar 24, 2015 at 15:34
• Yes, JBSregath is correct. Feel free to edit the puzzle (I am on a phone which makes editing very annoying.) Commented Mar 24, 2015 at 15:37