Assuming that you know about the Mine Sweeper game, find where mines can be in the following two scenarios:

1)
S S S S S
S 1 2 1 S
S X X X S

2)
S S S S S S
S 1 2 2 1 S
S X X X X S

Find for which 'n' values the following scenario is solvable for mines, where n represents the number of 2's.

S S S ........ S S S
S 1 2 ........ 2 1 S
S X X ........ X X S

In the above scenarios, S can be filled with a mine or a number and X can be filled only with a number.

1) 1m2m1    2) 01mm10     m = mine
   11211       012210
   00000       000000

Scenarios where 'n' is not a multiple of 3 appear solvable.

Where 'n' is a multiple of 3, there appears two
possible solutions where the mines may be:

3) 01mm2m1      1m2mm10
   0122211  or  1122210
   0000000      0000000

Where 'n' = 3k - 2, such that k is an integer value of 1 or greater, the top row of each scenario has a pattern of '1x1' with x being a number of 'm2m' for each k.

Where 'n' = 3k - 1, such that k is an integer value of 1 or greater, the top row of each scenario has a pattern of '01mxm10' with x being a number - 1 of 'm2m' for each k.

Where 'n' = 3k, such that k is an integer value of 1 or greater, the top row of each scenario has either a pattern of  '01mx1' OR '1xm10' with x being a number of  'm2m' for each k. 

By selecting either of the two end positions in the top or middle row, which are mine-free, the solvable pattern is revealed. Thus, ALL scenarios for 'n' are actually solvable.

Edited on February 16, 2009, 9:14 pm

Posted by Dej Mar on 2009-02-16 19:40:37