Hexominos
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What is a Hexomino?
Hexominos are figures you can form by six squares. - There are 35 different figures. Rectangles around Hexominos top
You can lay a rectangle as small as possible around a hexomino. Thus you have six groups. Rectangle Problem  top It is possible to form different rectangles with all figures with pentominos.
The problem is the same with hexominos. All 35 hexominos together have 35*6=210 squares. So there are many possible rectangles.

Example: Can you form a rectangle 21 x 10?

Statement: You cannot solve this rectangle problem.
Supposed there is a rectangle of all hexominos (see above). Then you colour the squares of the rectangle alternately and you get 105 white squares and 105 black ones. These are odd numbers.

On the other side you can colour the single hexominos. There are 24 "odd" hexominos with three black and three white squares and 11 "even" hexominos with four black and two white squares. (Two black squares and four white ones would also be possible.)
You can see that there are two ways to colour the hexominos. There are odd and even hexominos. In both cases you get an even number of coloured squares: 24*3, 11*4 or 11*2.

There is a contradiction: 105 is odd, the number of the coloured cubes is even.

Result: A rectangle 21*10 is impossible.

You can prove in the same way, too,  that there are no rectangles such as 14*15 or 6*35.

By the way you cannot build a rectangle with figures of  7 or more squares for another reason:
 ... There is one septomino with a hole. Here it is.

Figures of Hexominos top
 ... ... Compact rectangles of all hexominos are not possible.  Thus there are only rectangles with holes (light blue) or rectangles with some out-standing squares (red)  Two examples by Andrew Clarke. As is in the case of pentominos there are innumerable new figures formed of hexominos. On the left there is a parallelogram from Gardner's book (1). Figures of the Same Hexominos top You can build rectangles of the same hexominos (green). That isn't always possible.

There are many figures without rules (blue). It is nice you take the symmetry into account.

Special Hexominos top

Symmetric Hexominos 15 of 36 hexominos are symmetric. There are three groups, symmetric with one centre, one axis or two axes.

Nets of a Cube   top 11 Hexominos can be used as nets of a cube.

Hexominos with Rubik's Magic Master Edition  top Rubik's Magic is a folding game with the main problem: Transform a mat of 6x2 squares into a heart-shaped figure. Then five linked rings  change into five free rings on the reverse side.

If you fold the basic figure on the two symmetry axes, you get the simple rectangles 1x6 and 2x3. There are more complicated folding sequences, which lead to 16 hexominos. They are coloured in red below. It is remarkable that only odd pentominos develop. Hexacubes   top
 ... ... There are 166 figures of six cubes. There is an example on the left.

Hexominos on the Internet     top

Deutsch

Andrew Clarkes Poly Seiten
Polyominoes

Richard Mischak (Zahlenjagd)
Würfelpuzzles

English

Andrew Clarke's THE POLY PAGES
Polyominoes

gamepuzzles
Naming the HexominoesNaming the Hexacube Pieces

Gerard's Universal Polyomino Solver

Joseph Myers
Polyomino tiling

Steven Dutch
Polypolygon Tilings (archived)

Torsten Sillke
Tiling and Packing results

Wikipedia
Hexomino

References    top
(1) Martin Gardner, Mathematical Puzzles & Diversions, New York 1959

Gail from Oregon, thank you for supporting me in my translation.

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