Technique 4 of 5 · Hard boards
Pigeonhole
Confinement for groups. If two colors have all their open cells in the same two columns, those two columns' crowns belong to those two colors, and every other open cell in the two columns is ruled out.
The idea
Each color needs one crown and each column holds one crown. If two colors can only place their crowns in two particular columns, the two crowns fill those two columns, one each. There is no room left in those columns for any other color.
already ruled outthe cells the step looks atruled out by this stepa crown this step places
The example
On this 8×8 board, the blue and gray regions are small, and all of their remaining open cells lie in columns 6 and 7. Blue needs a crown and gray needs a crown, and both must land in column 6 or column 7: two crowns, two columns.
That uses up both columns. So the other open cells of columns 6 and 7, all eight of them in rows 1 to 4 (in the orange, teal and green regions), can't hold a crown. Cross them out. Notice that neither blue nor gray has a crown yet. You don't need to know where they will go to know where everything else can't.
The general rule
The same count works for any group size: three colors whose open cells fit in three rows own those three rows, and so on. It also works with lines and colors swapped: if the open cells of two rows all lie in two colors, those two colors' crowns are in those rows, and the rest of the two colors is ruled out. With one color and one line, it is simply Confinement.
How to spot it
- Look for small colors side by side. Two little colors next to each other along an edge are the classic case.
- Count. Pick a few colors and list the columns (or rows) their open cells use. If the number of lines equals the number of colors, you have a pigeonhole.
- Check the payoff. It only helps if some other color has open cells in those lines. Those are the cells you cross out.
Why it works
The name comes from the pigeonhole principle: two pigeons, two holes, one pigeon per hole, so both holes are full. Two crowns must fit in two columns with at most one per column, so each column's single crown is already spoken for.
In Logic Legends
The hint calls this step Pigeonhole and outlines the open cells of the colors involved. A board that needs it is graded Hard.