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Honey Islands

Be a bee!

In this honeycombs need to be divided into a given number of islands of a given size. Next to the puzzles is stated how many islands are required and what their size needs to be. Different islands never touch. Black hexagons are not part of any island. Some puzzles contain one or more bees. Each bee is part of one of the islands, but no two bees are ever part of the same island.

Example

In the puzzle to the left we need to find 3 islands that don't touch, that all exist of 3 hexagons that are connected to eachother. Maybe your visual insight will help you solve this puzzle, but we will show it can also be done by deduction.

Because some hexagons are black, just 13 white hexagons remain. Of those 9 belong to the islands we are looking for, which means there are exactly 4 white hexagons that need to be an obstacle.

Let's focus on the lowest 6 white hexagons in the 3 leftmost columns. They can not all be part of an island in the solution, because in that case those islands would touch. The same holds when you try to use any 5 of them. With 4 it is possible, but it can only be done in one way such that it is actually possible to make them into 2 islands that are large enough, but that possibility does not leave enough room for a third island in the upper right. From this we can deduce that at least 3 of these 6 hexagons need to be an obstacle, and thus at most 1 of all other white hexagons can be an obstacle.

Looking at those other 7 hexagons we note that not only is there at most 1 obstacle, there is also a need for at least one obstacle, because an island of size 7 or more is not allowed. There is even only 1 position where we can put this obstacle without creating an island of size at least 4! Having found where this obstacle needs to be, the rest of the puzzle has become trivial.


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