OFFSET
1,7
COMMENTS
Say that a tiling of a rectangle by other rectangles is prime if the only sub-rectangles in the tiling are those formed by a single tile. Say that two tilings are equivalent if there exists an inclusion/overlap-preserving bijection between the vertices, edges, and faces of every rectangle in them.
Problem 69 in Hugo Steinhaus's One Hundred Problems In Elementary Mathematics asks the reader to show that a(3) = a(4) = 0, and that there exist prime dissections for 5, 7, and 8 in which the pieces are of equal area. It cites Czesław Ryll-Nardzewski as proving that a(6) = 0, though this is not difficult to show by hand. The book also provides diagrams of both n = 7 solutions and four of the six n = 8 solutions.
Chung et al.'s paper Tiling Rectangles with Rectangles shows that the sequence grows at least as fast as c*2^(n/7) for some positive constant c, and states without proof that it is bounded above by 20000^n.
LINKS
F. R. K. Chung, E. N. Gilbert, R. L. Graham, J. B. Shearer, and J. H. van Lint, Tiling Rectangles with Rectangles, Mathematics Magazine, 1982.
/r/mathriddles, Distributing a rectangular inheritance.
Math StackExchange, How many "prime" rectangle tilings are there?
EXAMPLE
For n = 5 the a(5) = 1 example looks like
_____
| |___|
|_|_| |
|___|_|
.
For n = 7 the a(7) = 2 examples look like
_______ _______
| |_____| |_____| |
|_|___| | |___| | |
| |_|_| | |_|_|_|
|___|___| |_|_____|
CROSSREFS
KEYWORD
nonn,hard,more,nice
AUTHOR
Drake Thomas, Feb 01 2021
STATUS
approved