Numerous Solutions

Below is just one way to arrange the twelve pentominoes into a 6x10 rectangular shape. There are, in fact, 2339 different ways to do this.

Computers have made solving pentomino configurations nearly a trivial matter. Although it is still very enjoyable to solve such puzzles when purposely avoiding the help of machines, the internet can provide access to various pentomino solvers, for those so inclined.

One might fear, then, that there are few questions left unanswered concerning pentominoes.

Hardly. One simply needs to think of new (or tougher) questions.

Here's one. The above rectangle has 2339 different solutions. Is there some configuration that affords more answers?.

The answer is "yes", but here is an example of a question that would be almost pointless to ask unless access to computers was available. To try to find thousands of solutions by hand of a particular configuration would be ridiculously difficult. But with computer analysis examples can be found. The next figure has 3763 solutions:

Now, actually finding this particular configuration was not a trivial task. Determining how many solutions was easy enough, but how did I decide to test that shape in the first place? Well, I really just stumbled upon it while creating the Pentomino Daily Calendar (it is a figure entitled Letter "L", used for 25 July).

Another configuration produced even more solutions:

This shape (used on 15 March of the Calendar) has an amazing 5027 solutions.

Is there something with more?

Specifically, the following question can be posed: What configuration of 60 squares affords the most pentomino solutions?

The potential candidates don't even have to use fully connected squares. For example:

has 932 solutions (called, simply enough, 5x11Box, Plus One (V)). Of course, it is no where near the 5,000+ number from above.

Can someone provide a configuration with more solutions? Surely so. I suppose a computer program could be written to generate and test every potential candidate, but I'll leave that to someone else.

Believe it or not, I do have other things to do with my time.

If anyone does get higher numbers, please let me know, though; I'll be happy to post the results here.

- Eric Harshbarger, 5 January 2005

P.S.: Another interesting question related to this topic: can we find a sequence of configurations such that the number of solutions for the first is 1, the second is 2, the third 3, and so forth? How high can we proceed sequentially?

Good thing we have those computer solvers handy...

Update (14 Jan 2005): Aad van de Wetering of Holland sent me this list of figures, each with over 10,000 solutions (he did not guarantee this was a complete list):

7x9 with vertical domino at top left, monomino bottom right - 16720
8x8 with L-tetromino at a corner - 15512
7x9 with vertical domino at top left, monomino top right - 15482
8x8 with dominos on top two corners, one horizontal - 14791
7x9 with horizontal domino at top left, monomino top right - 14657
8x8 with L-tromino at a corner, adjacent corner - 14541
7x9 with horizontal domino at top left, monomino bottom right - 14191
7x9 with horizontal domino at top left, monomino bottom left - 13822
7x9 with L tromino at top left - 12945
7x9 with vertical domino at top left, monomino bottom left - 11381
7x9 with vertical tromino at top left - 10130
7x9 with horizontal tromino at top left - 10063

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