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The above puzzle is unsolvable. The following diagram shows the "solution" which has 0's in several cells where it is no possible to place a number.

*--------------------------*

| 4 2 6 | 1 5 9 | 3 7 0 |

| 7 0 5 | 2 3 6 | 4 1 9 |

| 3 1 9 | 8 7 4 | 6 2 5 |

|--------+-------+--------|

| 1 8 7 | 0 4 2 | 5 9 3 |

| 5 4 3 | 9 1 7 | 0 8 2 |

| 0 9 2 | 5 6 3 | 1 4 7 |

|--------+-------+--------|

| 2 3 4 | 6 0 8 | 7 5 1 |

| 6 7 1 | 4 9 5 | 2 3 0 |

| 0 5 0 | 3 2 1 | 9 6 8 |

*--------------------------*

You might have trouble believing this but I just looped through every possible value of all 40 blank cells subject to the given constraints. Now if there were no constraints that sounds like 7^ 40 steps, and even allowing that my

computer runs at about 140,000,000 steps/second, that would take something like 10^18 years. However the constraints reduce this dramatically, so that the whole process only took 470 seconds !

I just use visual basic in Excel, so lower level languages like C++ would no doubt be even faster. The trick is to pick your blank cell loops so that you can use a 30ish cell sum constraint as soon as possible.

I first came across this type of puzzle in a Sudoku forum that I am in, where someone could not solve one of these problems with the first and last rows being completely specified (in addition to the nine 30ish cells).

Well I just love a programming challenge and I looped through the 28 unspecified cells, and found the unique solution in less than perceptible time.

To find other similar puzzles I just did a google search and eventually found this thread, which naturally suggested the 40 blank cell problem which obviously would have multiple solutions.

An exhaustive search has revealed that the total number of solutions for the first puzzle (post 1) without any additional constraints is 19,879 and for the second puzzle (post 4) is 140,884.

Leren

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