Hashi:Linkdoku Bridges Puzzle ٤+

In Hashiwokakero(hashi, chopsticks, bridges) you are the king of the islands, and your job is to connect the islands according to their populations – small islands need only one bridge (it says 1 in the island), and larger islands need more bridges (the number in the island). Then remember that this is all one kingdom, yours, so you must make it possible for all the citizens to travel between all the islands. Also known as Bridges/Chopsticks



RULES:
1. Connect islands (the circles with numbers) with as many bridges as the number in the island.
2. There can be no more than two bridges between two islands.
3. Bridges cannot go across islands or other bridges.
4. The bridges will form a continuous link between all the islands.



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Solution methods:
Solving a Hashiwokakero puzzle is a matter of procedural force: having determined where a bridge must be placed, placing it there can eliminate other possible places for bridges, forcing the placement of another bridge, and so on.

An island showing '3' in a corner, '5' along the outside edge, or '7' anywhere must have at least one bridge radiating from it in each valid direction, for if one direction did not have a bridge, even if all other directions sported two bridges, not enough will have been placed. A '4' in a corner, '6' along the border, or '8' anywhere must have two bridges in each direction. This can be generalized as added bridges obstruct routes: a '3' that can only be travelled from vertically must have at least one bridge each for up and down, for example.

It is common practice to cross off or fill in islands whose bridge quota has been reached. In addition to reducing mistakes, this can also help locate potential "short circuits": keeping in mind that all islands must be connected by one network of bridges, a bridge that would create a closed network that no further bridges could be added to can only be permitted if it immediately yields the solution to the complete puzzle. The simplest example of this is two islands showing '1' aligned with each other; unless they are the only two islands in the puzzle, they cannot be connected by a bridge, as that would complete a network that cannot be added to, and would therefore force those two islands to be unreachable by any others.

Any bridge that would completely isolate a group of islands from another group would not be permitted, as one would then have two groups of islands that could not connect. This deduction, however, is not very commonly seen in Hashiwokakero puzzles.

Determining whether a Hashiwokakero puzzle has a solution is NP-complete, by a reduction from finding Hamiltonian cycles in integer-coordinate unit distance graphs. There is a solution using integer linear programming in the MathProg examples included in GLPK. A library of puzzles counting up to 400 islands as well as integer linear programming results are also reported.