Solving It - Part 2 of The Mathematics of Professor Alan's Puzzle Square
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The document provides step-by-step instructions for solving Professor Alan's puzzle square using commutator moves from group theory. It explains how to create clockwise and counterclockwise spins by combining column and row moves in different orders. This allows sorting the tiles into the correct rows and then using additional moves to rotate entire rows or subsets of tiles to put them in the proper order, solving the puzzle. It notes that solving the puzzle this way also provides an introduction to important mathematical concepts like abstraction and building complex operations from simpler ones.
Solving It - Part 2 of The Mathematics of Professor Alan's Puzzle Square
- 1. The Mathematics of Professor Alan's Puzzle Square part 2 – solving it https://magisoft.co.uk/alan/misc/game/game.html
- 2. How to do it? We are going to look at how to solve the puzzle square. It is not the fastest way, but is systematic and is based on a mathematical concept from Group Theory called a commutator
- 3. Don’t panic! Group Theory is something that is studied at universities, so this sounds scary but: 1. You don’t need to understand it to follow the solution method. 2. Actually it is not as scary as all that, and we’ll see later how a little bit of maths can help solve this and similar puzzles.
- 4. First steps Try pressing the arrows in the order here. 1. Second column up 2. Top row right 3. Second column down 4. Top row left 1 2 3 4
- 5. A little cycle See what has happened. Almost everything is in the same place, but we took three tiles for a little anti- clockwise spin. 1 2 3 4
- 6. A long word for it In maths speak we have just used a commutator. This means: 1. Do something 2. Do something else 3. Do the opposite of step 1 4. Do the opposite of step 2 1 2 3 4
- 7. Spin the other way If we want the three tiles to spin round clockwise, we can simply press the opposite of each arrows in the opposite order … Reset the square and try it. 1 2 3 4 reset
- 8. Two useful moves We now have two sets of moves that cycle the tiles focused on the 2 square. As the bottom two rows aren’t affected, we’ll focus on the top two rows for now 1 2 3 4 1 2 3 4
- 9. Half way there! Using these moves, in fact just one of them, we can almost solve the puzzle. We’ll just aim to get the right tiles in the right rows. Like this square.
- 10. Let’s do it Here’s a pretty scrambled square. We don’t need our fancy commutator moves to get the top row right. First rotate a few columns up or down to get it roughly there. Just 2 to go.
- 11. Row 1 Just tile 2 to go. Shift it to the clear column. Then move it up to the top row. … and now for the second row
- 12. Row 2 5 and 8 are OK, so just 6 and 7 to go. 6 is almost there, we just do an anti-clockwise spin. Note this is focused on the tile on the last column and second row, so you need to adapt the move.
- 13. Different focus tile Try this out on an unscrambled square to see how to do it. It is just the same, you just choose the arrows on the row and column you want to spin around. 1 2 3 4
- 14. Row 2 continued Just tile 7 to go. We could do this by moving it across one followed by two spins.
- 15. Jumping rows However, we can save time by moving several rows at once. Do a spin move, but press the up/down arrows twice. Here it is on a reset square. 1 2 36 4 5
- 16. Nearly there For the last two rows, just follow the same pattern, of shift and rotate. Here we just need to swop tiles 11 and 14 and they are already in the right position for a clockwise spin.
- 17. Half way done … or maybe better All the tiles are in the right rows, so we just need to sort out the row order. For the coloured puzzle square the news is even better. All the tiles in the same row are identical anyway, so you are done 1 3 1 1 1 0 1 2 1 5 1 6 1 4 4 78 21 9 6 3 5
- 18. A right hand cycle We saw how to create both clockwise and anticlockwise spins, but both have triangles pointing left. By simply doing left first for the column we can create a right pointing triangle. 2 3 4 1
- 19. Shuffle a row Now we can combine the left and right pointing triangles. The moves the 6 to the top row and the moves it back down. In the end only the top row tiles are moved. + =
- 20. Last steps We can now work on a single row at a time, rotating the whole row, or just three tiles with our new move. The shuffled top row is now easy to fix, as is the last row.
- 21. Last steps The middle rows are a bit more difficult. The top row we can shuffle the whole row two steps left. The last row we can rotate the last three tiles.
- 22. Stuck? In both of these rows, it is just the last two tiles that are the wrong way round. If this were a 5x5 or a 3x3 square, then we might be stuck (we’ll see why in a later part), but as rotates all four and rotates three, we can combine them.
- 23. A simple swop In both of these rows, it is just the last two tiles that are the wrong way round. If this were a 5x5 or a 3x3 square, then we might be stuck (we’ll see why in a later part), but as rotates all four right and rotates three left, we can combine them.
- 24. Solved So, we can just do this to the two middle rows … … and we are done ✔
- 25. … and as well as solving it … Yes, more maths 1. We’ve had our first taste of commutators 2. We’ve focused on moves rather then the tiles … this is a form of abstraction critical in both maths and computing 3. We’ve packaged up bigger moves built out of simpler moves … this is crucial for algebra in maths and it is also at the heart of programming