Cracking Minesweeper with Z3 SMT Solver

For those who are not very good at playing Minesweeper (like me), it's possible to predict bombs' placement without touching debugger.

Here is a clicked somewhere and I see revealed empty cells and cells with known number of "neighbours":

What we have here, actually? Hidden cells, empty cells (where bombs are not present), and empty cells with numbers, which shows how many bombs are placed nearby. Here is what we can do: we will try to place a bomb to all possible hidden cells and ask Z3 SMT solver, if it can disprove the very fact that the bomb can be placed there.

#!/usr/bin/python

WIDTH=9
HEIGHT=9

known=[
"01?10001?",
"01?100011",
"011100000",
"000000000",
"111110011",
"????1001?",
"????3101?",
"?????211?",
"?????????"]

from z3 import *
import sys

def chk_bomb(row, col):

    s=Solver()

    cells=[[Int('cell_r=%d_c=%d' % (r,c)) for c in range(WIDTH+2)] for r in range(HEIGHT+2)]

    # make border
    for c in range(WIDTH+2):
        s.add(cells[0][c]==0)
        s.add(cells[HEIGHT+1][c]==0)
    for r in range(HEIGHT+2):
        s.add(cells[r][0]==0)
        s.add(cells[r][WIDTH+1]==0)

    for r in range(1,HEIGHT+1):
        for c in range(1,WIDTH+1):

            t=known[r-1][c-1]
            if t in "0123456789":
                s.add(cells[r][c]==0)
                # we need empty border so the following expression would be able to work for all possible cases:
                s.add(cells[r-1][c-1] + cells[r-1][c] + cells[r-1][c+1] + cells[r][c-1] + cells[r][c+1] + cells[r+1][c-1] + cells[r+1][c] + cells[r+1][c+1]==int(t))

    # place bomb:
    s.add(cells[row][col]==1)

    result=str(s.check())
    if result=="unsat":
        print "row=%d col=%d, unsat!" % (row, col)

for r in range(1,HEIGHT+1):
    for c in range(1,WIDTH+1):
        if known[r-1][c-1]=="?":
            chk_bomb(r, c)

The code is almost self-explanatory. We need border for the same reason, why Conway's "Game of Life" implementations also has border. Whenever we know that the cell is free of bomb, we put zero there. Whenever we know number of neighbours, we add a constraint, again, just like in "Game of Life": number of neighbours must be equal to the number we got from the Minesweeper. Then we place bomb somewhere and check.

Let's run:

row=1 col=3, unsat!
row=6 col=2, unsat!
row=6 col=3, unsat!
row=7 col=4, unsat!
row=7 col=9, unsat!
row=8 col=9, unsat!

These are cells where I can click safely, so I did:

Now we have more information, so we update input:

known=[
"01110001?",
"01?100011",
"011100000",
"000000000",
"111110011",
"?11?1001?",
"???331011",
"?????2110",
"???????10"]

I run it again:

row=7 col=1, unsat!
row=7 col=2, unsat!
row=7 col=3, unsat!
row=8 col=3, unsat!
row=9 col=5, unsat!
row=9 col=6, unsat!

I click on these cells again:

I update it again:

known=[
"01110001?",
"01?100011",
"011100000",
"000000000",
"111110011",
"?11?1001?",
"222331011",
"??2??2110",
"????22?10"]

row=8 col=2, unsat!
row=9 col=4, unsat!

This is last update:

known=[
"01110001?",
"01?100011",
"011100000",
"000000000",
"111110011",
"?11?1001?",
"222331011",
"?22??2110",
"???322?10"]

... last result:

row=9 col=1, unsat!
row=9 col=2, unsat!

Voila!

The source code: https://github.com/dennis714/yurichev.com/blob/master/blog/minesweeper/minesweeper_solver.py.

More Z3 examples are in my blog and here: https://yurichev.com/tmp/SAT_SMT_DRAFT.pdf.

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The page last updated on 05-March-2017