Pentomino Relationships

Introduction Examples 3 by 20 4 by 15 5 by 12 6 by 10 Links Tools Used

I found my first Pentanome solution on Boxing Day 1982, a day and a half of intermittent playing after Dad had given them to me.

There is something very nice about the way the twelve pieces of wood, each a different shape and each made of five cubes fit together into rectangles or cuboids, but if that was all there was to them, I would soon have become bored with them.

The puzzle really begins after you've found a rectangle or cuboid, and the challenge then is to find as many other solutions as you can from the one you've just found, and it is very likely that others can be "seen" if you look hard enough.

There are just 17 of the 368 4x15 solutions, 63 of the 1010 5x12 solutions and 358 of the 2339 solutions from which I can't "see" any other solutions.

This site contains all the rectangular solutions grouped into families, and for each family, I have tried to show how the different members are related by drawing on selected members.

It would also be possible to merge families of different sized rectangles, though I haven't done so here. The most obvious example is given by the family of 16 5x12 solutions and the family of 33 6x10 solutions which contain solutions with two separate 5x6 rectangles.

I also have some very nice 3x4x5 cuboid solutions which I might add later, if I can find a nice way of representing them. Some of the "moves" associated with these solutions are very nice.

In my student years, I was fascinated by the fact that the puzzle is hard to solve but once you've solved it, you can often chain together a number of "moves" to get a family of solutions. I felt like it was trying to tell me something. Finding solutions became a bedtime routine and the number of "moves" I could see grew as I compared the new solutions to the one I had found already.

It feels a bit strange to think that I've spent so much time on something which didn't have as much to tell me as I thought it might, and that I've looked hard at every rectangular solution, but the puzzle has certainly been an excellent source of entertainment and computing excercises for me, and I hope this web page, my last pentanome exercise (I hope), is of interest to somebody.

RELATIONSHIP EXAMPLES

I have allowed a number of different type of relationship between solutions, some of which are easier to spot than others.

The first example shows that both 3x20 solutions are related by a subshape which has rotational symmetry.

Another example of rotational symetry is:

There are also subshapes with reflexive symmetry as in:

and similiar subshapes which can be swapped, as in:

Sometimes you see two subshapes each made with two pieces which aren't similiar, but the pieces from one can be used to make the other. I don't believe that I have used this type of relationship though because where I have seen it, there has been another chain of moves connecting the solutions.
Then there are the subshapes which can be made in a different way with the same pieces as in:

I have allowed these subshapes to have up to three movable components as in:

and

The latter of these can be thought of as removing the F like shape and putting it back somewhere else.
This type of relationship is the hardest to see and might be thought of as cheating, but I allow them because I can see them, even if I do have to look quite hard sometimes.

FAMILIES OF SOLUTIONS

I have already shown the 3 by 20 solutions.

4 by 15
17 singles.   (10 at time)
23 sets of 2.   (10 at time)
9 sets of 3.
16 sets of 4.   (10 at time)
7 sets of 6.
2 sets of 7.
2 sets of 8.
1 set of 10.
3 sets of 12.
1 set of 14.
1 set of 20.
1 set of 22.
1 set of 40.

5 by 12
63 singles.   (10 at time)
67 sets of 2.   (10 at time)
18 sets of 3.   (10 at time)
22 sets of 4.   (10 at time)
7 sets of 5.
10 sets of 6.
3 sets of 7.
3 sets of 8.
5 sets of 9.
4 sets of 10.
3 sets of 11.
3 sets of 14.
1 set of 15.
4 sets of 16.
1 set of 17.
1 set of 19.
1 set of 20.
1 set of 23.
2 sets of 25.
1 set of 29.
1 set of 30.
1 set of 41.
1 set of 63.

6 by 10
358 singles.   (10 at time)
216 sets of 2.   (10 at time)
63 sets of 3.   (10 at time)
65 sets of 4.   (10 at time)
17 sets of 5.   (10 at time)
25 sets of 6.   (10 at time)
9 sets of 7.
12 sets of 8.
7 sets of 9.
8 sets of 10.
4 sets of 12.
2 sets of 13.
2 sets of 14.
1 set of 15.
2 sets of 16.
1 set of 17.
2 sets of 18.
1 set of 20.
1 set of 21.
1 set of 29
1 set of 32.
1 set of 33.
1 set of 36.
1 set of 37.
1 set of 45.
1 set of 50.
1 set of 58.

Links

There are quite a lot of web pages out there with Pentomino content, and many of them can found from the few links I've included here.
Both Anna's page and Jay Jenicek's page have ideas for making you own and Jay's page is also a good source of other things to do with them. Anna's picture of her home made Sculpey set (a member of the 6x10 set of 33) and her description of her home made agate set make my plain brown wooden ones look very dull.
If you are going to make your own out of a block of something solid, it might be worth spending some time choosing a solution to use as a pattern. Depending on the cutting tool you've got and your source material, a 3x20 solution might be a good choice, or you could use one of the last 2 16 member families of 5x12 solutions and start with 2 5x6 blocks or a 5x7 and a 5x5 block.
Frank Ruskey's page is a good source of links, and while Eric Weisstein's page doesn't have a lot on Pentominoes, it is an excellent source for anything remotely mathematical.

TOOLS USED

These images and web pages were made with:
1) A C program to find all the solutions.
2) A C program with a Wish(TCL/TK) interface to display the solutions -
described in [Matt Welsh, Linux Journal, issue 10, Feb 1995].
3) 'xud | xwdtopnm | ppmtogif' to grab the displayed solutions to gif files.
4) xpaint to draw on selected solutions.
5) vi and bash to generate the html pages.