# Polyform Exclusion, Equalization, Variegation, and Integration

No room! No room! they cried out when they saw Alice coming.
—Lewis Carroll, Alice's Adventures in Wonderland

The exclusion problem is to remove as few cells as possible from a given region of the grid so as to exclude a given polyform. Equalization problems involve equalizing the distribution of cells within a region. The variegation problem is to color the cells of the grid with as few colors as possible so that a given polyform, no matter where it lies on the grid, has cells of all different colors. Integration means joining copies of a weakly joined polyform to make a strongly joined polyform.

## Exclusion

 Hexomino Exclusion. Exclude a hexomino from a checkerboard. Polyking Exclusion. Exclude a polyking from a checkerboard. Polyiamond Exclusion. Exclude a polyiamond from the plane. Polyhex Exclusion. Exclude a polyhex from the plane. Polycairo Exclusion. Exclude a polycairo from the plane.

## Equalization

 Connected Polyomino Magic Squares. Arrange adjacent copies of a pentomino in a square grid to place the same number of cells in each row and column. Two-Pentomino Magic Squares. Arrange copies of two pentominoes in a square grid to place the same number of cells in each row and column. Polyabolo Magic Squares. Arrange copies of a polyabolo in a square grid to place the same number of cells in each row and column.

## Variegation

 Polyiamond Variegation. Color the cells of the polyiamond plane so no copy of a polyiamond has duplicate colors. Polyhex Variegation. Color the cells of the polyhex plane so no copy of a polyhex has duplicate colors.

## Integration

 Polyking Integration. Join copies of a polyking to form a polyomino.

Back to Polyform Curiosities.
Col. George Sicherman [ HOME | MAIL ]