Let us consider the space as the number 9, and look at the a picture in the normal order. 9 is 123456789, its reverse order number is 0, which is even arrangement; b picture is 123456879, reverse order number is 1, it is odd arrangement. We know that we are only able to move 9, and the movement here is equivalent to a special swap. Now suppose that from the b-picture through a series of translations to the a-picture, the space block 9 must be moved (changed) an even number of times (the first time to the left must return to the right again, the next time must be back again) According to the above theorem, the resulting sequence must be an odd arrangement (same as the b picture), whereas the a picture is evenly arranged, thus creating a contradiction, so the b picture cannot be transformed into a final a picture by translating the space block.

　　Further consideration, a picture can be translated into some other state, we classify these into one category, b picture also represents a class, now ask "how many types of puzzles?", the answer is more than or equal to 2 * 2 puzzle There are only two of these categories. Here is only the proof:

　　1. According to the above theorem, all puzzles are divided into at least two categories.

　　2. There are only two types of 2*2 puzzles.

　　3. After the puzzle is enlarged, the number of classifications does not increase.

　　According to these three articles, it can be concluded that there are only two types of puzzles. We can come up with some other interesting results. The difference between the two types of puzzles is that they have an odd number of swaps, that is, if you exchange a puzzle for a non-empty plate for an odd number of times, it will change to another category.

　　The essential reason for dividing into two categories is because the plane has two faces, just like clockwise and counterclockwise.

variant

　　The pictures of the puzzles are mostly based on natural scenery, buildings and patterns that are familiar to people. Castles and mountains are two traditional themes, but any picture and image can be used as a puzzle. Some companies also offer services that make private photography work into puzzles.

　　Common flat puzzles are available in 300, 500, 750, and 1,000 pieces; currently the largest size puzzle is 24,000 pieces/set. The mainstream puzzle layout, the number of pieces/set is 38×27, totaling 1,026 pieces; 500 pieces/set is 27×19, 513 pieces. The size and number of pieces of children's puzzles are more varied.

　　There are also some kinds of jigsaw puzzles with patterns printed on both sides. Players can group them on any side of the pattern, and the difficulty of the game is also increased. Because the pieces in the hand are difficult to determine which side is the player. The right side.

　　In addition to the traditional flat puzzle type of three-dimensional puzzle. The pieces are made of a more solid material such as wood or foam. The spatial characteristics often lead to an increase in difficulty, requiring the player to stitch the pieces in a specific order: if the finished part has a small piece of improper assembly, the remaining pieces are likely to continue to splicing up.

　　Another type of toy called a "puzzle box" is also common: players can assemble small drawers or boxes with intermediate storage items using interlockable parts like flat puzzle pieces.

　　It can be seen as an intermediate type between a flat puzzle and a three-dimensional puzzle. Similar to the plane jigsaw puzzle, its spherical surface is also a single-layer structure with a piece of cardboard; and its final shape is a three-dimensional shape with long, wide and high properties. Spherical puzzle

　　In addition, there are some software that can simulate physical puzzles in a computer graphics. The advantage of this virtual puzzle is that there is no need to manually clean it, and there is no risk of losing pieces. Among them, metamorphosis puzzles or famous animated character puzzles are more popular.