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Sliding block puzzle

By David Shaw, 10 February 2016

Take 12 dominos and arrange them into a rectangle.

You will need

• Set of dominoes
• Blu-tack
• Sticky labels, pens, scissors (optional)

What to do

1. Turn all the dominos face down. You won’t need to look at the dots in this activity. Arrange them so that the shorter side is towards you.
2. Arrange 12 dominoes in a rectangle three dominoes wide and four dominoes long.
3. Get some more dominoes, and Blu-tack them to the table around the outside of this rectangle to make a frame. Leave a small gap of around 2 mm around the rectangle, to give the pieces room to move.
   

   Put a frame around, and then remove the dominos.

4. Take the dominoes out from inside the border you just made.
5. Blu-tack two dominoes together end-to-end. Try not to use a lot of Blu-tack – you want the dominoes to be as close together as possible. Put these dominoes inside the frame, in the top left corner. Put a sticky label on this shape.
6. Blu-tack two dominoes side-by-side to make a square. Put these in the bottom left corner of the frame, and then put one more domino to the right of it to complete the bottom row.
7. Blu-tack two dominoes side-by-side to make a second square. Put these in the bottom right of the space remaining in the frame, and then put one more domino to the left to complete the row
8. Put two dominoes in the remaining space inside the frame, leaving an empty space for two dominos. The puzzle should now look something like this:

The aim of the puzzle is to get the long thin tile in the top left corner down to the bottom right corner without picking it up. To do this you can slide the pieces up, down, left or right. You can’t take pieces out of the frame and put them back in, and you can’t rotate the pieces either.

What’s happening?

You can stick paper on top of the blocks to make them easier to see.

The puzzle you have just made is a type of sliding block puzzle. These puzzles have been around for over one hundred years, and used to be very popular amusements before the invention of television. One of the most famous is called 15 and requires you to rearrange fifteen numbered blocks to put them in order. More modern sliding block puzzles have pieces of a picture on each block, and the goal is to re-create the whole picture.

The puzzle you just made is known as Hughes’ Puzzle. It has two dominoes missing from a full rectangle. To move the tall skinny piece sideways, you need to have both spaces one on top of the other, and to move the square pieces up or down, you’ll need to have the two spaces side by side.

Real-life maths

Try to get the long rectangle to the far corner!

There are many strategies you can use to solve a sliding block puzzle. However, a trick that works on one sliding puzzle might not work on another. For some sliding block puzzles, the best way to find the answer might not be much better than just randomly sliding blocks until it has been solved.

Some mathematicians study how long it would take a computer to answer a question. They put each type of question into a category based on how hard it is. Sliding block puzzles are in the category PSPACE-complete, which is a very hard category. Sliding block puzzles take more time for a computer to solve than Sudoku, but they aren’t as hard as a game of chess.

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