Mapping cubes with holes onto cubes with handles

Tiling with notched cubes

In 1966, Golomb showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In 3 dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a ‘notched cube’) that tiles a larger copy...

Connectivity of Fibonacci cubes, Lucas cubes, and generalized cubes

If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd( ↽Ð f ) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the grap...

Sums of cubes with shifts

Zero Knowledge with Rubik's Cubes

Since the invention of the Rubik’s cube by Ernö Rubik in 1974, similar puzzles have been produced, with various number of faces or stickers. We can use these toys to define several problems in computer science, such as go from one state of the puzzle to another one. In this paper, we will classify some of these problems based on the classic Rubik’s cube or on generalized Rubik’s Cube. And we wi...

Hyper-tubes of hyper-cubes

Hyper-tubes consisting of hyper-cubes of n-dimensions were designed and formulas for substructures of vary dimensions established.