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Platform

Archibald the Wise has so many books that they don't fit in his bookcase anymore, so he wants to build a new one. For this task he bought a bunch of shelves, and chains and posts with which he can attach the shelves to the ceiling, the floor or other shelves. Of course this bookcase needs to be stable.

The numbers next to the diagram indicate how many shelf sections are in their row. A shelf consists of one or more shelf sections and needs to be attached to a chain or a post on each end. The numbers above the diagram indicate how many chain sections are in their columns; the numbers below the diagram tell how many post sections are in their columns.

A chain can be used to attach a shelf to the shelf above, provided that that one is already stable, or to the ceiling. A post can be used to a attach a shelf to the shelf below, provided that it is already stable, or to the floor. Shelves need to be supported on both ends by being attached at their endpoints, and there will be no posts or chains that are redundant because they are not used to attach a shelf.

Example

In this example in the first column there need to be two chain sections. Those have to be in the upper two positions, because the floor can never be hanging from a chain. For the same reason the two post sections in the fifth column must be in the bottom two positions. In the second column the chain has to be in the highest position and the post in the lowest, otherwise there would be a shelf that was both supported by a post and hanging from a chain on a single end.

All chains and posts need to be functional, so there must be a shelf hanging from the chain in the first column. That shelf has a width of just one unit, because it is supported on the other end by the post in the second column. For the same reason, there has to be a shelf on the post in the fifth column. This one needs to be at least two units in length, or the chain in the second column couldn't be functional anymore.

The chain in the fourth column can not be hanging from the ceiling anymore now, but needs to be in the middle. This chain needs to be functional, so there is a shelf section either to the left or to the right of it. To the right it could never be stable, so it is on the left. The shelf can be at most one unit in width, given that there are only two shelf sections on this row. The left side still needs to be attached, and there is only one chain available for that.

Finally, the upper shelf needs to have a width of three to be able to have its left side attached, and that solves the puzzle.

Background

This genre was invented by Maarten Löffler.


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