# What's new

## Unchequer the board

By David, 27 April 2017 Activity

Here’s a quick sliding puzzle to get your brain pumping. See if you can separate the light and dark counters in as few moves as possible!

#### You will need

• Pen and paper
• 8 dark counters
• 7 light counters

#### Setting up

1. Draw a 4 x 4 grid of squares

Start by drawing a square grid, four squares wide and four squares tall. Make sure the squares are larger than your counters.

2. Make a chequerboard pattern with the counters. Start by putting a light counter in the top left square. Then put a dark counter in the next square down, then a light, and finally a dark counter to finish the column.

Arrange the counters in a chequerboard pattern.

3. For the second column, start with a dark counter and alternate colours as you move down the column.
4. Repeat this procedure, starting the third column with a light counter and the final column with a dark counter. The final square is left empty.
5. You now have a chequerboard pattern of light and dark counters on your grid.

#### The challenge

1. As you slide the counters, the chequer pattern will disappear

The aim is to separate the counters, so all the black counters are on one side of the grid, and all the white counters on the other side.

2. Move counters by sliding. There’s one empty square in your grid. You can move a counter up, down, left or right into the empty square from an adjacent square.

When the white counters are on one side and the black on the other, you’re finished!

3. Now a different square is empty. For your next move, slide a counter into the new empty square.
4. Keep sliding counters around until all the light counters are on one side, and all the dark ones are on the other.
5. Once you’ve finished this puzzle, have another go and count how many moves you took. What’s the fewest number of moves you can finish in?

#### What’s happening?

This type of puzzle is known as a sliding block puzzle. One nice thing is that it’s always possible to solve. No matter what moves you make, you can always find a solution. If you think you’ve really messed up, you can always take all your moves backwards and get back to the start.

The Maths by Email team don’t know the quickest answer, but we can estimate. If you remove all the light counters from the board, it takes eight moves to get the dark counters to one side. If you’re just using the light counters it’s a bit quicker – it only takes five moves to get to the other side. So the solution must take at least 13 moves. We can further estimate that it will take twice that many moves, as half the moves will be opposing up and down, or side to side. So a solution of 26 moves sounds plausible.

After several attempts, our best solution took 27 moves. We think this is pretty good, but there might be a better solution out there. If you find one, let us know!

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