1. Under this circumstance, which 2 squares would you open?

Getting the answer requires you to “simulate” the mines: i.e. predict all the possible locations of the mines. Let’s do this pictorially, where black spots represent possible mine locations and black lines mean that a single mine can be in any of those squares.

This is the first possibility. Let’s call it A1.

(click to enlarge each picture)

The second possibility is here, let’s call it A2.

Let’s now look at an impossibility:

No mine can be placed that satisfies the 2, 3 and 4 conditions at the same time.

A1 and A2 are the only possible outcomes, and with those all combined, this is what you get:

The two unfilled squares on the second column are hence the ones that can be opened safely.

2. If, out of Q1, the sum of the numbers in the 2 squares is 6, what would you do?

This is not a difficult question if you managed to pass the first step, but some simulation might be needed. There are 6 unsolved mines on the field, and in the first question, there are 3-4 possible mines identified, but you have a 50-50 chance of getting them right if you don’t know the numbers in the 2 squares you open.

Assume A1. Let’s simulate a situation when the sum of both numbers is 6:

I’ve colour coded the opened squares according to the colours on the field: the top one is a ‘5’ and the bottom one is a ‘1’. Since you already have 2/5 of the mines identified for the number on top, the 3 other possible spots (in the 1st column) must all be mines. However, this would mean that there are a total of 7 unmarked mines in the game! Clearly this is not possible.

Let’s look at A2 instead.

This, unlike the previous scenario, would be possible. Now consider other possible combinations for the 2 squares that will add up to 6: you have 3/3, 4/2 and 5/1. Either way, you will be able to solve the puzzle, because there are squares that you can definitely open. The squares that you can open for sure now are here, all marked in gold:

3. If, out of Q1, the sum of the numbers in the 2 squares is 3, what would you conclude?

The way to solve this question is basically the same as the previous question. Let’s assume A1 again.

The lines marked in gold are safe zones – there are no possible mines there because if the sum of the numbers in the 2 squares is 3, . If you now count the total number of mines that you can mark, you have 5, 1 short of the stipulated 6. If we assume A2, you’d have 4 mines, 2 short of 6. This means that if the sum of the numbers in the 2 squares is 3, your puzzle is unsolvable – trick question!

4. Which squares in the 1st column are the safest to open from the current position? Which is the most useful square to open out of these squares?

Basically, safest to open = lowest probability of being a mine. Useful square = if you do not hit a mine, you will be able to gather the most information from the square you guess.

Let’s look at safety first. There is approximately a 50% chance of A1 happening, meaning that there’s approximately the same chance of A2 happening. If A1 is the layout, there is a one in third chance that you will hit a mine if you guess any of the bottom three squares in the first column. If A2 is the layout, you will not hit a mine by guessing any of the bottom three squares in the first column.

There are either 3 or 2 mines in the first column excluding the bottom 3 squares, and there are 7 squares in total above . If there are 2 mines (A1), there’s a 2/7 chance of hitting a mine – if there are 3 (A2), it’s a 3/7 chance.

This means that the total chance of hitting a mine for the bottom three squares is 1/3 (0ne in three) x 1/2 (50% chance of A1 happening). Total chance of hitting a mine for the top 7 squares is (2/7 + 3/7)/2. This means that its safer to open the bottom three squares.

How about usefulness? Let’s assume you open the last square in the first column. The number would either be 3 or 4, depending if the square above it was a mine or not. The last square would thus be useful iff you got a 4 because you’d be able to mark a mine and open the other 2 squares around the ‘3’ in the second column (and etc etc). If you got a 3, you’d be able to open the second last square, but that would be all.

How about the second last square in the first column? If you opened it, you’d definitely get a 1 (or a mine). That’s because the ‘3’ in the second column only requires 1 more mine. You wouldn’t be able to proceed further if you guessed the second last square.

If we open the third last square, the number would be 1, 2 or 3. If it were 3, you’d know that the square above and below the third last square, as well as the square diagonally to the right of the third last square would be mines. This would be very useful. If it were 2, you can only hazard a guess – the 2 mines could possibly have two placements (won’t bother to describe them). If it were 1, it’d be safe to open the fourth last square in the first column, which might allow you to open other squares.

It follows that either the last square or the third last square is the most useful, depending on what you get when you open the squares – but it seems more likely that the third last square would be useful to open.