• http://www.frankwester.net/mine_standard.html

• in each example, assume undrawn neighboring squares are numbers (not affecting the count)
• a - in an example means 'don't care' - in other words, a numbered square whose value doesn't affect the squares being shown

Heuristics:

1. (Easy) if a numbered square's neighboring exposed mine squares number the same as the number, expose all unexposed neighbors. In this example we can expose a, b, c and d.

X	2	a
X	3	b
c	d	X

The next sections require you to generate sets of 'must be true' (MBT) and 'must be false' (MBF statements for each numbered square, e.g....

1	a
2	b
2'	c
3	d
X	e

@	MBT	MBF
1	{A or B}	{A and B}
2	{A and B} or {A and C} or {B and C}	{A and B and C}
2'	{B and C} or {B and D} or {C and D}	{B and C and D}
3	{C and D} or {C and E} or {D and E}	{ C and D and E}

Note, at 2' we really have {B and C and `D}, {B and D and `C}, {C and D and `B} where ` means 'not'

2. Walk through the MBT statements for the entire puzzle. If two 'or' MBT's differ by a letter, then you can eliminate that letter. This is like finding a least common denominator. For example here we can eliminate 'c' (but we can't know whether it's a or b)...

1	a
2	b
X	c

3. Compare all the MBT statements with all the MBF statements. If any match, you can eliminate them. For example, we can eliminate {A and B}...

1	a
2	b
2'	c
3	d
X	e

@	MBT	MBF
1	{A or B}	{A and B}
2	{A and B} or {A and C} or {B and C}	{A and B and C}
2'	{B and C} or {B and D} or {C and D}	{B and C and D}
3	{C and D} or {C and E} or {D and E}	{ C and D and E}

---

Logic examples

1.

1	X	X	X	2
3	5	a	4	2
X	X	b	3	X
X		c	X	2

@	proves
5	a or b
4	a or b
3	a or b or c

Therefore, 'c' has no mine.

2.

1	X	X	X	2
3	5	a	4	2
X	X	b	3	X
X		3'	X	2

@	proves
5	a or b
4	a or b
3'	b or c

Therefore, inconclusive - nothing to do.

3.

X	2	1	1
X	4	2	X
a	b	c	2

@	proves
4	{a and b} or {a and c} or {b and c}
2	b or c

Therefore,

• we can eliminate {b and c} - it contradicts b or c
• we know we must have 'a'; {a and b} or {a and c} == {a} + {b or c}
• stuck! Indeterminate!

4. TODO: indeterminate example where you compute the odds and take the best case

-- MattWalsh - 22 Feb 2002