It’s colorful. It’s playful. It’s mathematical. It’s artistic, even linguistic. It’s about shapes. Sometimes it’s easy; sometimes it’s hard. It’s brain training or it’s pure fun. It’s sure to have intergalactic appeal.
What is it?
It’s a game called IPTU.
If you take identical squares and arrange them so that they share at least one edge you can make shapes called polyominoes. If you choose five squares you can make the following 12 different arrangements. (Rotations and reflections don’t count.)
These are called Pentominoes (ref. 1). By convention they have been assigned letters of the alphabet. If you take the 12 Pentominoes and fit them together they will fit perfectly into a 6 x 10 rectangle (as well a many other rectangles and shapes).
Even more amazing, they can be arranged in this 6 x 10 rectangle 2339 different ways!
With 2339 different ways available, surely it would be easy to find one. It’s not. It’s surprisingly difficult, even frustratingly difficult, but I would recommend you try it sometime. When you succeed it’s a mini triumph.
When you don’t succeed, this is what frequently happens. You’re cruising along and everything’s fine. You’re saying to yourself, I’m pretty clever. Any dummy could do this. Then you have one more space to be filled and … uh-oh! That shape has already been used.
When you encounter this kind of impasse, you don’t reach for your hammer. It won’t help. You have to remove some shapes and try again.
But how many do you remove? This can take thousands of trials and it still might not work. (In this case IPTU knows that you could remove the last two and make it work.)
Sometimes you can be in real trouble. For example, suppose in your youthful enthusiasm and innocence you place 4 shapes on your 6 x 10 tray:
Pretty proud of yourself so far? Well, don’t be. You’re doomed. You can spend a million years and you won’t fit the other shapes in. This is a dead end — really dead. How do I know? Well …
Even Pentominoes devotees have found only a small fraction of the 2339 solutions without using a computer, and that’s no fun. Also, most of the rarer and most interesting solutions are unlikely to be found without help — for example, the eleven solutions with the I shape completely buried in the interior.
So, what can you do?
IPTU uses a compromise:
1. Let the computer do the dull work of filtering out the many thousands (or millions) of non-solutions.
2. Provide a series of clues to guide players to any one of the thousands of interesting solutions via game cards.
There are many different ways that clues can be provided in IPTU, and the resulting difficulty can vary enormously.
In this example the order of placement of pieces on the perimeter has been given and, to make it even easier, the X shape has been placed, and the asterisk (*) even tells you where to start! The player starts by placing the first shape ( U ), so that it covers the asterisk (*). Here, the U could be oriented any of 4 ways:
The player then places the rest of the shapes in the order given, proceeding in the direction of the arrows (counterclockwise). Sometimes you’ll find it easier to go clockwise. It doesn’t matter.
At the bottom of the card are the remaining Pentominoes in the interior of the 6 x 10 matrix. Just remember, IPTU in this example shows you the order of shapes to be placed, not the orientation.
Pentomino playing pieces — cardboard, plastic, wood, tiles, concrete blocks, … and a 6 x 10 tray in which to place them are very helpful for more challenging puzzles.
The solution to the above puzzle is:
In IPTU fitting the Pentominoes pieces is only the beginning. Within the completed 6 x 10 matrix are smaller groups of Pentominoes (nuggets) I’ll call flips, shufls, swaps, rects, … which the human (or alien) mind must find. They are pattern-recognition and spatial challenges (ref. 2).
These nuggets are named by taking the letters representing the shapes and alphabetizing them. As you might expect (or fear) these nugget names are not unique, hence the names NPVZ(2) and NP(2).
NPUVXZ, and IL are flips. They are symmetrical nuggets which can be removed from the 6 x 10 matrix, flipped, and replaced in the same space. They then form a new solution.
Flipping IL creates FLTY which is a rect, a rectangular nugget. It’s very special because it can flipped or reflected in four ways:
NPVZ(2) is a shufl. It is not symmetrical but has the fascinating property that the Pentominoes making up the shape can be rearranged two or more ways. Once NPVZ(2) is shuffled, it contains another flip, NP(2).
Finding the solution, flips, rects, shufls, and swaps as well as other nuggets could earn you points, whether you’re playing alone or competitively. Of course, you can’t claim a nugget unless you name it.
Pronouncing the names of the nuggets you’ve found, e.g., IL, FLTY, IPTU, LUVZ, or even NPUVXZ, can be a requirement in competition. It’s not for the faint-hearted and can be a linguistic challenge, but the attempts can also be fun in small-group play.
To purchase an IPTU starter set (6 x 10 tray, 12 different Pentominoes, and 10 Level 1 game cards) go to Our Products.
1. Golomb, Samuel W., Polyominoes, Princeton University Press, 1994.
2. Bhat, Rashmi and Audrey Fletcher, Pentominoes, http://www.andrews.edu/~calkins/math/pentos.htm, 1995.