How to assemble a Chinese cube from 6 parts. Wooden puzzle knots made from bars

Date of: 2013-11-07

The world is designed in such a way that things in it can live longer than people, have different names at different times and in different countries, we can even play The Simpsons games. The toy you see in the picture is known in our country as the “Admiral Makarov puzzle.” In other countries it has other names, of which the most common are “devil’s cross” and “devil’s knot”.

This knot is connected from 6 square bars. The bars have grooves, thanks to which it is possible to cross the bars in the center of the knot. One of the bars does not have grooves; it is inserted into the assembly last, and when disassembled, it is removed first.

The author of this puzzle is unknown. It appeared many centuries ago in China. In the Leningrad Museum of Anthropology and Ethnography named after. Peter the Great, known as the “Kunstkamera”, there is an ancient sandalwood box from India, in the 8 corners of which the intersections of the frame bars form 8 puzzles. In the Middle Ages, sailors and merchants, warriors and diplomats amused themselves with such puzzles and at the same time carried them around the world. Admiral Makarov, who visited China twice before his last trip and death in Port Arthur, brought the toy to St. Petersburg, where it became fashionable in secular salons. The puzzle also penetrated into the depths of Russia through other roads. It is known that the devil’s bundle was brought to the village of Olsufyevo, Bryansk region, by a soldier returning from the Russian-Turkish war.

Nowadays you can buy a puzzle in a store, but it’s more pleasant to make it yourself. The most suitable size of bars for a homemade structure: 6x2x2 cm.

Variety of damn knots

Before the beginning of our century, over several hundred years of the toy’s existence, more than a hundred variants of the puzzle were invented in China, Mongolia and India, differing in the configuration of the cutouts in the bars. But two options remain the most popular. The one shown in Figure 1 is quite easy to solve; just make it. This is the design used in the ancient Indian box. The bars of Figure 2 are used to create a puzzle called the “Devil’s Knot.” As you might guess, it got its name due to the difficulty of solving it.

Rice. 1 The simplest version of the "devil's knot" puzzle

In Europe, where, starting from the end of the last century, the “Devil's Knot” became widely known, enthusiasts began to invent and make sets of bars with different cutout configurations. One of the most successful sets allows you to get 159 puzzles and consists of 20 bars of 18 types. Although all the nodes are externally indistinguishable, they are arranged completely differently inside.

Rice. 2 "Admiral Makarov's Puzzle"

The Bulgarian artist, Professor Petr Chukhovski, the author of many bizarre and beautiful wooden knots from different numbers of bars, also worked on the “Devil's Knot” puzzle. He developed a set of bar configurations and explored all possible combinations of 6 bars for one simple subset of it.

The most persistent of all in such searches was the Dutch mathematics professor Van de Boer, who with his own hands made a set of several hundred bars and compiled tables showing how to assemble 2906 variants of knots.

This was in the 60s, and in 1978, the American mathematician Bill Cutler wrote a computer program and, using exhaustive search, determined that there were 119,979 variants of a 6-piece puzzle, differing from each other in combinations of protrusions and depressions in the bars, as well as placement bars, provided that there are no voids inside the assembly.

Surprisingly large number for such a small toy! Therefore, a computer was needed to solve the problem.

How a computer solves puzzles?

Of course, not like a person, but not in some magical way either. The computer solves puzzles (and other problems) according to a program; programs are written by programmers. They write as they please, but in a way that the computer can understand. How does a computer manipulate wooden blocks?

We will assume that we have a set of 369 bars, differing from each other in the configurations of the protrusions (this set was first determined by Van de Boer). Descriptions of these bars must be entered into the computer. The minimum cutout (or protrusion) in a block is a cube with an edge equal to 0.5 of the thickness of the block. Let's call it a unit cube. The whole block contains 24 such cubes (Figure 1). In the computer, for each block, a “small” array of 6x2x2=24 numbers is created. A block with cutouts is specified by a sequence of 0s and 1s in a “small” array: 0 corresponds to a cutout cube, 1 to a whole one. Each of the "small" arrays has its own number (from 1 to 369). Each of them can be assigned a number from 1 to 6, corresponding to the position of the block inside the puzzle.

Let's move on to the puzzle now. Let's imagine that it fits inside a cube measuring 8x8x8. In a computer, this cube corresponds to a “large” array consisting of 8x8x8 = 512 number cells. Placing a certain block inside a cube means filling the corresponding cells of the “large” array with numbers equal to the number of the given block.

Comparing 6 “small” arrays and the main one, the computer (i.e., the program) seems to add 6 bars together. Based on the results of adding numbers, it determines how many and what kind of “empty”, “filled” and “overflowing” cells were formed in the main array. “Empty” cells correspond to empty space inside the puzzle, “filled” cells correspond to protrusions in the bars, and “crowded” cells correspond to an attempt to connect two single cubes together, which, of course, is prohibited. Such a comparison is made many times, not only with different bars, but also taking into account their turns, the places they occupy in the “cross”, etc.

As a result, those options are selected that do not have empty or overfilled cells. To solve this problem, a “large” array of 6x6x6 cells would be sufficient. It turns out, however, that there are combinations of bars that completely fill the internal volume of the puzzle, but it is impossible to disassemble them. Therefore, the program must be able to check the assembly for the possibility of disassembly. For this purpose, Cutler took an 8x8x8 array, although its dimensions may not be sufficient to test all cases.

It is filled with information about a specific version of the puzzle. Inside the array, the program tries to “move” the bars, that is, it moves parts of the bar with dimensions of 2x2x6 cells in the “large” array. The movement occurs by 1 cell in each of 6 directions, parallel to the axes of the puzzle. The results of those 6 attempts in which no “overfilled” cells are formed are remembered as the starting positions for the next six attempts. As a result, a tree of all possible movements is built until one block completely leaves the main array or, after all attempts, “overfilled” cells remain, which corresponds to an option that cannot be disassembled.

This is how 119,979 variants of the “Devil’s Knot” were obtained on a computer, including not 108, as the ancients believed, but 6402 variants, having 1 whole block without cuts.

Supernode

Let us note that Cutler refused to study the general problem - when the node also contains internal voids. In this case, the number of nodes from 6 bars increases greatly and the exhaustive search required to find feasible solutions becomes unrealistic even for a modern computer. But as we will see now, the most interesting and difficult puzzles are contained precisely in the general case - disassembling the puzzle can then be made far from trivial.

Due to the presence of voids, it becomes possible to move several bars sequentially before one can be completely separated. A moving block unhooks some bars, allows the movement of the next block, and simultaneously engages other bars.

The more manipulations you need to do when disassembling, the more interesting and difficult the puzzle version. The grooves in the bars are arranged so cleverly that finding a solution resembles wandering through a dark labyrinth, in which you constantly come across walls or dead ends. This type of knot undoubtedly deserves a new name; we'll call it a "supernode". A measure of the complexity of a superknot is the number of movements of individual bars that must be made before the first element is separated from the puzzle.

We don't know who came up with the first supernode. The most famous (and most difficult to solve) are two superknots: the “Bill's thorn” of difficulty 5, invented by W. Cutler, and the “Dubois superknot” of difficulty 7. Until now, it was believed that the degree of difficulty 7 could hardly be surpassed. However, the first author of this article managed to improve the "Dubois knot" and increase the complexity to 9, and then, using some new ideas, get superknots with complexity 10, 11 and 12. But the number 13 remains insurmountable. Maybe the number 12 is the biggest difficulty of a supernode?

Supernode solution

To provide drawings of such difficult puzzles as superknots and not reveal their secrets would be too cruel to even puzzle experts. We will give the solution to superknots in a compact, algebraic form.

Before disassembling, we take the puzzle and orient it so that the part numbers correspond to Figure 1. The disassembly sequence is written down as a combination of numbers and letters. The numbers indicate the numbers of the bars, the letters indicate the direction of movement in accordance with the coordinate system shown in Figures 3 and 4. A line above a letter means movement in the negative direction of the coordinate axis. One step is to move the block 1/2 of its width. When a block moves two steps at once, its movement is written in brackets with an exponent of 2. If several parts that are interlocked are moved at once, then their numbers are enclosed in brackets, for example (1, 3, 6) x. The separation of the block from the puzzle is indicated by a vertical arrow.

Let us now give examples of the best supernodes.

W. Cutler's puzzle ("Bill's thorn")

It consists of parts 1, 2, 3, 4, 5, 6, shown in Figure 3. An algorithm for solving it is also given there. It is curious that the journal Scientific American (1985, No. 10) gives another version of this puzzle and reports that “Bill's thorn” has a unique solution. The difference between the options is in just one block: parts 2 and 2 B in Figure 3.

Rice. 3 "Bill's Thorn", developed using a computer.

Due to the fact that part 2 B contains fewer cuts than part 2, it is not possible to insert it into the “Bill’s thorn” using the algorithm indicated in Figure 3. It remains to be assumed that the puzzle from Scientific American is assembled in some other way.

If this is the case and we assemble it, then after that we can replace part 2 B with part 2, since the latter takes up less volume than 2 B. As a result, we will get the second solution to the puzzle. But “Bill’s thorn” has a unique solution, and only one conclusion can be drawn from our contradiction: in the second version there was an error in the drawing.

A similar mistake was made in another publication (J. Slocum, J. Botermans “Puzzles old and new”, 1986), but in a different block (detail 6 C in Figure 3). What was it like for those readers who tried, and perhaps are still trying, to solve these puzzles?