## Solution - Quantum Minesweeper

by Derek Kisman

This puzzle is just what it says on the tin: A Minesweeper game following Quantum Mechanical rules. The basics can be picked up from randomly clicking around. The “quantum state” of the 15x15 board is a uniform superposition of many possible “classical states”, each consisting of 100 mines placed around the board. In the classical version clicking on a square with a mine would kill you, but in the quantum version, it simply restricts the superposition to only those states with no mine on that square. The exception is if all remaining states have a mine on that square, in which case you die. In all possible Universes. Ouch.

The numbers in clicked squares work similarly to the classical version, but instead of counting the number of adjacent mines, they count the average number of adjacent mines across all remaining states. Thus, they tend to not be integers. Also, clicking somewhere on the board tends to decrease the immediately adjacent numbers (since you've removed any possibility of a mine in that square), but increase the numbers everywhere else, since the remaining mines are now distributed among fewer possible squares. In practice, this mostly means you can click around freely until the last 100 squares, at which point they are guaranteed to kill you. And indeed if the starting superposition consisted of every possible configuration, this is how it would work.

But you may notice that this isn't quite true; there are a few irregularities. Indeed, from experimentation you might discover that clicking on certain squares on the board will strongly affect other, distant, squares (much more than the small increase you'd expect). Ah, this is another quantum mechanical feature: some squares are “entangled” with each other! The first step is to identify exactly which squares have entanglements. You can determine (with varying levels of difficulty) that the entanglements are partitioned into thirteen sets of four squares:

Each square in a set behaves indistinguishably from the others, but still the entanglements don't all behave the same way. Figuring out their exact nature is a little tricky, and might require saturating the rest of the board then experimenting.

As it turns out, each set allows its constituent squares to contain a number of mines from some subset of {0, 1, 2, 3, 4}.

1. {4} (i.e. every square in this set is guaranteed to contain a mine)
2. {4, 2, 0}
3. {4, 1} (ie, either every square has a mine or exactly one square has a mine)
4. {3, 0}
5. {4, 2}
6. {4, 3, 0}
7. {3, 0}
8. {3, 2, 1}
9. {2}
10. {2, 0}
11. {0} (no squares in this set ever contain a mine)
12. {4, 2}
13. {3}

The bottom row provides a sort order for the sets. Interpreting the configurations as binary bits (i.e. {4} = 2^4), the answer to this puzzle can be read out: PURITY IN DEATH. Now try not to think of how many of your many-worlds alternate selves were purified while solving this puzzle…