Similar Hole Problems

In 1974 Jean Meeus proposed the problem 'to construct an area containing an interior hole of the same shape' for sets of polyominoes and polyiamonds. If the total area of the set of pieces is A then we require to find integers x, y and n such that A = n(x²-y²) in which case we have a pattern of area nx² with a similar hole of area ny². It is also possible to look for multiple congruent figures with similar holes.

The links below lead to various sections of the site where solutions to this problem may be found.

Polyominoes	Polyiamonds
Pentominoes	One-sided hexiamonds
One-sided pentominoes	Heptiamonds
Hexominoes	Enneiamonds
One-sided Hexominoes

 

Polycubes	Polyaboloes
	Pentaboloes
	One-sided Pentaboloes

 

Polyares
Tetrares
One-sided triares

 

Other Polyforms
Domsliced Pentominoes	One-sided strip hexominoes
One-sided domsliced tetrominoes	One-sided tridominoes
One-sided 2-4 cellominoes	Domsliced 3½-ominoes