# Blog

## Swap the blocks

By

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Here’s a puzzle that’s easy to set up, but surprisingly tricky to solve. See if you can swap two counters just by sliding them!

## You will need

• Paper
• Textas
• 1 red counter
• 1 blue counter
• 3 white counters

Tip: If you don’t have any counters, you can use small pieces of paper and colour them with textas.

## Setup

1. On your paper, draw a rectangle about 15 centimetres wide and 10 centimetres tall.
2. Draw a line across the middle of the rectangle, cutting it in two.
3. Draw 2 lines down the rectangle, dividing it into 6 squares. Don’t worry if the squares aren’t perfect!
4. Colour the top left square red.
5. Colour the bottom left square blue.
6. Put a blue counter in the red square and a red counter in the blue square.
7. Leave the top right square empty, and put a white counter in the remaining three squares.

## The game

1. The goal of the game is to have the blue counter in the blue box and the red counter in the red box at the same time.
2. To make a move, slide a counter into an empty box from a neighbouring box.
3. Counters can only go up, down, left or right. They can’t go diagonally, and they can’t jump over other counters.
4. After you’ve done it a few times, try counting your moves. See how small you can get that number!

## What’s happening?

Since there’s only one empty space, the counters seem to follow each other around the board. The next move will put a counter in the spot that just got emptied!

When the empty space is in a corner, you only have two options, and one of them is to undo your last move! The most important moves are when the gap is in the middle 2 squares.

You might notice that the counters seem to move in circles. You can get them to circle around in the 4 squares on the left, or the 4 squares on the right, or a big circle around all 6 squares.

You might notice the red and blue counters following each other. To solve the puzzle, you’ll need to swap the order that they are in. You might want to split them apart and then move them back together in a different way.

If you solve the puzzle but don’t know how, try again. After a few goes, you can try counting your moves to see how quickly you can do it. Anything under 30 moves is pretty good. Our top score was just 18 moves!

## Real-life maths

If you find sliding puzzles really hard, you’re not alone! Mathematicians categorise the puzzles as PSPACE-complete, which means that even computers can find it challenging to come up with a solution. 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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## One response

1. Martin

Once you know the trick to swapping the two counters, sliding puzzles become really easy. I usually solve all but the last two rows of a sliding puzzle and then slowly use this technique to get the last two rows solved together.

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SPOILER ALERT – The 18 move solution (maybe one of the 18 moves solutions?) is as follows

The following directions tell you which counter to move. For example, if the letter is “R” then move a counter to the right – since there’s only one counter that can move right, then that’s the one to move.

R, R, U, L, L, D, R, U, R, D, L, L, U, R, D, R, U, L

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