JP2019013723A - Three-dimensional magic square toy - Google Patents

Abstract

To provide a three-dimensional magic square toy that children can take in their hands, fiddle around, and contemplate.SOLUTION: Numbered belts 2, 3, 4 corresponding to degrees are wound around a support 1 formed in a hexagonal column shape for the number of degrees (in this case, three) successively from the belt 2, and a player turns the belts 2, 3, 4 each manually to pry out a correct magic square having six faces for the hexagon while confirming whether the correct magic square is formed.SELECTED DRAWING: Figure 1

Description

The present invention relates to an educational toy using a magic square.

Magic squares are mathematics and play completed by arranging numbers in a vertical and horizontal plane, and the sum of the vertical, horizontal, and diagonal numbers is the same. Of the magic squares that have been known to the world, the more advanced ones can be said to be the work of the people who are doing professionally even if they look at books and the world. Most of the magic squares that ordinary people know is a single plane, and it often ends in a simple way. At present, there are only three-dimensional magic squares to count. It is natural that if the degree increases and the number increases, it will become difficult to understand, but we hope that we will deepen our understanding and become good at calculation at the child stage. The following are representative examples of plane and solid objects.

Prior patent documents

Patent Literature

The World of Magic Square Kiyomi Omori Japan Critics Company (2013) Mathematical puzzle encyclopedia Tomio Ueno Meishodo (2016)

Problems to be solved by the invention

For those who propose this, it is unfortunate that even though it is a three-dimensional object, it is a three-dimensional object on paper, which children do not pick up and play around with. If you examine it, you can see that it is a well-established knowledge and the world, even if this is not all, and you can see the struggle. In the book, the solid magic square is mainly the sum of the solid diagonals, and the sum of the planes (the sum) does not matter so much, but as a proponent, it is not so particular as an ordinary person. Intend. The solid diagonal is the sum of the numbers on the line that connects the corners and corners that pass through the center of the dice in a state where numbers are densely (layered) in the magic square that is in the shape of a dice. Although it is difficult to see and confirm this number directly, it is a result derived only by calculation, so what the proposer has put out as a polyhedron magic square is, It looks similar, but the contents are a little different.
Simple works can have the same solid diagonal even with a calculator, but full-scale works are difficult to complete without the help of a computer. Therefore, I think that works made with the help of these computers are beyond the scope of toys and will be left to specialists. There is also a concern that the work will be limited if you stick to the solid diagonal. Although it is natural that a difficult work is more valuable, it is not a world that is generally not well known and easy to understand without hints, but the results of researchers and enthusiasts working hard historically We want to convey to children mainly fun and surprise. In order to make this, it is necessary to learn not only the decimal system but also the calculation method according to the degree such as the quaternary system and the quinary system, and I would like to introduce that various works including fractions can be made.

Object of the present invention Creation of a three-dimensional magic square toy for children to pick up and play around.

Means for achieving the object 1) As means for achieving this, it is considered to use a belt or a tape. In other words, not only people can see and show the completed three-dimensional magic square, but they also have to judge whether it is a magic square by seeing the numbers arranged by themselves. I want to achieve the goal in the direction of thinking and making it.
2) For example, in the form of a toy, a belt that has a number of figures corresponding to the order of the magic square on the outside of a solid pentagon or hexagon (more than a triangle) frame. Is wound around the number of orders.
3) For example, if it is a hexagon, a hexahedron magic square is written, and if it is a triangle, a trihedron magic square is written on the belt, but this belt can be turned freely by humans. The correct magic square appears only when the belts of the order number are placed in the correct positions.
4) For example, there are extra meaningless numbers on the belt, so even if one side of the magic square is completed, the other side of the magic side will collapse so that it will not be completed. Yes. This is also because the empty numbers are written irregularly, but if the empty numbers are written regularly, all six sides of the hexahedron are completed as magic squares at the same time. Of course it is possible to make this so that you can choose either. Because it is irregular, when you complete one side and try to complete the next side, the completed side will collapse. The belt itself and the main body are all molded in one piece, so it is inconvenient to carry, so we devised it so that it can be divided and manufactured.However, considering the fact that it is still bulky, consider the next idea. It was.
5) For example, in the method of making this polyhedron, as the degree increases, the hexagon is simply increased to 7 and 8 and it is inconvenient to handle. In other words, you can imagine what is shaped like a cassette tape, and stack the numbered cassette tape by the next number. In this case, the belt does not have to be long like the belt described at the beginning, and the numbers are wound around the reel, so the overall number will not increase regardless of the number of faces. Moreover, it can make as many faces as the tape allows, and the belt type needs to match the length to the support, but the reel type does not. And if you use both sides of the tape, you can double the amount on the tape. The part number can be entered on the reel. The advantage of simpler manufacturing and the cover can also be used to hold the tape. Moreover, if you look at a small number with a magnifier so that you can turn the reel without removing the cover, it would be an accessory or earring, and it would seem cute.
6) For example, the tape is kinked if it is too thin, so it must be strong enough not to kink. Pulling out the tape is done by a person turning the reel that is wound around directly, but when it is returned, it is returned to the opposite reel by turning the opposite reel.
7) For example, in order to prevent the reels from turning easily, something like a felt felt or something should be attached. Or is it possible to make the reel shafts directly brake each other by stacking cassettes?
8) For example, a slide-type (detachable) transparent plate is provided on the front of the cassette to prevent it from popping out, including the prevention of tape bending. If possible, a lens with a lens effect is better because it wants to make the numbers appear larger.
9) For example, it is the same in entering empty numbers, but both can be changed to a magic square with a different arrangement by simply exchanging belts and tapes. Also, the same belt and tape, but the NO.1 belt and tape are in a symmetrical position (NO.1 and NO.4 in the case of the fourth order). You can also use the magic square property of becoming a magic square.
10) Next, for example, the pyramid-type magic square is a stack of individual blocks with numbers in the shape of a magic square. This example is a four-stage pyramid that combines 10th to 4th orders, but this can be further combined with a 12th-order pyramid to form a 5-stage pyramid. The feature is that another pyramid magic square is hidden in the pyramid. Of course, it is not so difficult to increase the number of steps, and it is also possible to hide not only the large numbers but also the pyramids where the numbers become smaller by taking the form of fractions smaller than 1, 2 It is also possible to make three or more pyramids.
11) For example, I would like to propose the creation of software for making these magic squares.
12) For example, for a circle, the total in the circumferential direction does not change by rounding the sphere shape, but the radial numbers can be separated, and the radial numbers can be combined. If the background is the Earth or a human face, the magic square can be completed by breaking the earth or human face. If the background is a decent picture or shape, the magic square is disjointed. It is possible to create an original work by bringing in a picture of your favorite person, and it is now possible to create a magic square on the face of a three-dimensional sliced person instead of a photograph. Another way is to create as many pins or frames as you need, and insert them into a predetermined place. Here, the numbers in the circumferential direction are also matched. In this case, it is not always necessary to cut the ring, but if the foundation is a human face, it can be used together. Although it may be a matter of judgment whether to stab a pin in a person's face, there is nothing wrong with thinking the same way as acupuncture. Other spheres can be combined with dolls by making the sphere look like a barbell, and can be tailored into a doll with a barbell. It is possible to conceive not only a die-shaped type but also various shapes by connecting multiple circles.
13) As with all things, miniatures can be used as accessories and souvenirs.
14) For example, another three-dimensional magic square, but with the aim of making it easier to see, the idea is to first make six holes in the sphere and then insert a thin stick with both sides tapered to assemble the magic square. . The holes are at an angle of 90 degrees, top and bottom, left and right, and front and back. The hole of the sphere is tapered, and the angle of the taper is adjusted so that it stops only when the rod is inserted. Numbers are marked on the sphere at three locations, upper and lower, so that the numbers can be checked regardless of the direction of the sphere. It is unsuitable for imagining and confirming a solid diagonal line when there are dense blocks of magic squares. Therefore, we try to make the numbers in the magic square visible by passing a thin stick between the numbers. Many books have such expressions, but the results can be confirmed directly with the eyes. If this type of 3D magic square is a child who is good at mental arithmetic, it would be possible to assemble it if you get used to it without seeing the answer. It may be possible to compete for assembly speed, and it may be surprisingly effective to test the limits of the human brain. It's easier to feel because you don't have to imagine the results on paper. If you have a ball and a connecting rod, you can increase the order and the visual effect seems to be enough. By the way, the number of connecting rods is 54 for the 3rd order, 144 for the 4th order and 300 for the 5th order. This type can be used not only for solid objects, but also for normal planar magic squares. Since the number of spheres is 125 in the fifth order, it is possible to correspond to the eleventh magic square if only the plane is considered. Furthermore, by making the numbers written on the sphere correspond to Braille, the numbers can be understood by touching them, and it can be a toy for people who are blind.
15) For example, it is a 3D magic square in another form, but it is the same as the previous section in that it is easy to see. Let it form. The solid diagonal magic square made up of 27 numbers is united in a block shape, and 27 sets that are different little by little are added. Starting from the beginning is difficult for beginners, so the numbers in the block are pre-filled from 1 to 729 to form a magic square from the beginning. The completed block magic squares can be freely combined in 27 sets to create a 9x9x9 large diagonal diagonal magic square. This uses 27 sets of small block magic squares and tries to create a larger composite diagonal magic square.

1) Expanding the regularity and possibilities of numbers.
2) Children's interest and mastery of arithmetic operations.
3) Expansion of three-dimensional expression and understanding through visual and tactile sensations that do not rely solely on books.
4) Development of attention to numbers and perseverance.
5) The joy and sense of accomplishment of creating Guinness-class things without special talent.

Perspective view of 3D magic square polyhedron Sectional view and plan view Fig. 1 (Position relationship of belts in which all six sides are completed as magic squares) Belt position relationship diagram 2 (Position relationship of belts that have been completed as magic squares with “upper side” on the first side and “lower side” on the second side only) Fig. 3 (Position of the belt completed as a magic square on the upper side and the lower side of the fourth side) Belt position relationship diagram 4 (Position relationship of the belt completed as a magic square for “upper side” on the fifth side and “lower side” on the sixth side) Belt diagram and cover diagram Perspective view of solid magic square polyhedral tape type 1 Front view Perspective view of solid magic square polyhedral tape type 2 Front view and cover drawing Front view and cover drawing of polyhedral tape type 3 Solid magic square polyhedron completed figure 1 (hexahedral) Table of the completed drawing 2 (hexahedral) Table of completed figure 3 (hexahedral odd order) Polyhedral pyramid perspective view and layout (4 stage solid block magic square peripheral expansion type) Another perspective view of the pyramid type (pyramid type parent-child magic square) Pyramid plan view Same pyramid type even completed chart (Pyramid type parent and child magic square even type completed form) Pyramid type odd number completed chart (Pyramid type parent and child magic square odd type completed type) Same pyramid completed chart 3D Magic Square Polyhedron Ball Connected Perspective View Completion chart (5x5 3D Magic Square ball connection type complete form) Perspective view and side view of solid magic square polyhedron composite type 1 Another perspective view of the composite type 1 Perspective view of the composite type 2 3D magic square polyhedron circle front view and plan view Table of the completed figure 1 (sphere type circle-triple type magic square) Completion chart of another three-dimensional magic square polyhedron (sphere type circle-quadruple type magic square) A perspective view and a plan view of the three-dimensional magic square polyhedral block 1 A perspective view and a plan view of another solid magic square polyhedral block 2 Complete fraction type chart 1 (pentahedral magic square fraction type complete form) Completion chart 2 of the same fraction type (divided by numbers of 1 to 9 by pentahedral magic square 3x3, 4x4, 5x5 type-this is the total number and then back-calculated to each fraction (It is a fitted form) Solid magic square polyhedron exchange type even-parent configuration 1 Configuration 2 Configuration diagram 3 Solid magic square polyhedron exchange type odd-parent configuration 1 Configuration 2 Solid magic square polyhedral belt type wheel type configuration diagram

The three-dimensional magic square polyhedron according to the first invention will be described with reference to FIGS. 1 to 7. The number of the three-dimensional magic square polyhedron corresponding to the number of orders entered in the hexagonal columnar support 1 as shown in FIG. The belts 2, 3 and 4 are wound around the order (in this case, 3) in order from 2, and the player turns each of the belts 2, 3 and 4 by hand to check whether they are in the correct magic square. However, because it is hexagonal, it is a game that will seek out the correct magic square with six faces. Each belt 2, 3 and 4 is held by being sandwiched by a flange portion 8 protruding below the support 1 and an upper lid 6 which is placed on top of the flange portion 8, so that each belt 2, 3 and 4 is slippery. An inner ring 5 is provided between the support body 1 and the support body 1. The inner ring 5 is adhered to a thin film 12 one by one (for the number of surfaces) so as to easily follow the hexagonal shape, and is adapted to the dimensions of the outer periphery of the support 1. Since the support body 1 is bulky when it is made as one body, the support body 1 is divided and the six surfaces are connected by pins 9. The bottom of the support 1 has six recesses 10 in order to keep the hexagon securely, and a notch is formed between the bottom lid 7 and the convex portion 11 of the bottom lid 7 to engage with the top surface of the bottom lid 7. The lower end of the support 1 is received at the outer periphery, and the hexagon is maintained and stabilized. Therefore, it is designed to be separated when the pins 9 connecting the supports 1 are pulled out. It is to be noted that the lower lid 7 is desired to have a slight taper on the recess 10 and the protrusion 11 so that the lower lid 7 does not fall easily. Although the inside of the support body 1 is hollow, this is because a small thing of the same shape is put into a thing with a large degree, and it is considered that a matryoshka state is brought about. Next, belts 2, 3 and 4, but as shown in FIGS. 3 to 6, each belt 2, 3 and 4 has a meaningless empty number (alphabet part, ie, ab aa. bb.aaa.bbb) are regularly or irregularly arranged, and this empty number does not constitute a magic square. The player tries to find the correct magic square by turning these belts 2, 3 and 4 by hand to check if they are in the correct magic square. Since the empty numbers are irregularly arranged, once you complete only one correct magic square, if you try to complete the next magic square, the already completed magic square will collapse. ing. In this way, it will be a play that continues to make magic squares until you know the arrangement with your head. The reason for this is explained in FIGS. As a feature of this play, there is a characteristic peculiar to the magic square that the characteristic as the magic square is maintained even if the upper belt 2 and the lower belt 4 are exchanged. However, be careful because none of them are possible. The belts 2, 3 and 4 are provided with small projections 13 between the numbers so that the belts 2, 3 and 4 can be easily turned. Material, hardness, and length are important factors, so we need to be careful. Although the numbers used are consecutive numbers, there is plenty of room for selection, such as even numbers only or odd numbers only. In addition, the number of faces and numbers can be increased without limit to hexahedrons, so the magic squares are rich in variety. Basically, any number of prisms can be used as long as the order is triangular or more. The total number (definite sum) changes depending on how to use the numbers to be used and empty numbers, but if the total is shown, the answer will be simpler, so the total will not be shown. In addition, the cover 14 was put on the entire hexagon so that the unnecessary numbers could not be seen so that only the necessary magic square numbers could be seen. When the belt is turned, it is necessary to remove the cover 14 temporarily. However, if the user gets used to it, the cover 14 is considered unnecessary. Although the scales of the cover of FIG. 1 and the cover of FIG. 7 are different, it should be understood that the window is actually enlarged in the direction of rotation because empty numbers are not represented in FIG. When the belt 2, 3, 4 is configured as shown in FIG. 3, magic squares can be formed simultaneously on six sides. Here, the empty numbers are regularly arranged. When the belts 2, 3, and 4 are configured as shown in FIGS. 4, 5, and 6, any one of the six faces can form one of the magic squares. Here, empty numbers are arranged irregularly. Depending on how you choose the empty numbers, you can complete only two or three faces at the same time.

The second invention takes into consideration that the first invention is bulky in size. Since the explanation regarding the contents as the magic square is the same, this feature will be omitted, and this feature will be described with reference to FIGS. As shown in FIG. 8, the present invention intends to change from a belt type to a tape type. The cassette tape-shaped support 15 as shown in FIG. 8 is inserted into the notches 23 of the support 15 by the projections 24 at both ends of the reel 20 from both sides, and from one reel 20 to the front surface 15a of the support 15. It is assumed that tapes 16, 17 and 18 are provided which are drawn out along the other reel. The support 15 is overlapped by the same number of orders, and the lengths of the tapes 16, 17, and 18 on which the magic squares are written are determined by the number of numbers used. It will be exactly the same as the invention. Since there is no restriction on the length of the tape, it is possible to provide a magic square with a long arrangement as long as the tape allows. The reels 20 at both ends are turned by hand, and the magic squares of the correct arrangement of 3 steps (for the order) are selected on the front surface of the cassette. Grooves 21 are cut in the upper and lower parts of the front surface of the support 15 so that a cover 19 that serves as a tape presser is attached with a lens effect for the purpose of preventing the tape from bending and for those who are hard to see numbers. . The tape can move between the front surface 15 a of the support 15 and the cover 19. The cover 19 is attached by being inserted from the side so that the upper and lower protrusions can slide into the groove 21. The tapes 16, 17, and 18 are put in and out by hand, but the reels are slightly put out on both sides of the support body 15 so that it can be easily turned. Replacement of the tape 16, 17, 18 can be easily done by sliding the tape cover 19 and removing it from the groove 21 on the front side, and only the necessary part of the cover 19 is transparent. The numbers that seem to get in the way are made opaque so you can't see them. Also, to prevent the reel from turning easily, it is necessary to provide a felt-like resistance to increase the frictional force against the reel, and the tape itself must be strong enough to prevent the tape from twisting. There is. Further, a positioning structure is provided by providing irregularities 22 so that the positions of the supports 15 do not shift when the supports 15 are stacked.

The third invention proposes another form by judging that the second invention is not structurally simple yet. As shown in FIGS. 10 and 11, the reel and tape are the same as in the second aspect of the invention, but the reel 26 and the tapes 27, 28 and 29 rotate to the two poles 32 protruding from the support 25. As much as possible, the respective parts overlap each other directly, and a cover 30 that also serves as a tape presser covers it. This cover 30 is the same as the second invention described above in that it has a transparent portion and an opaque portion so that the cover 30 prevents the tape from being bent and the numbers that are likely to get in the way are not visible. The replacement of the reel 26 is easy if the cover 30 is removed, and the reel 26 can be directly rotated from the outside of the notch 33 of the cover 30, so that the complexity of the parts can be eliminated. There are gaps through which the tapes 27, 28, 29 pass between the cover 30 and the front and back surfaces of the tape base 25a of the support 25, and numbers are entered using both the back and front of the tapes 27, 28, 29. If this is the case, another number arrangement can be enjoyed simply by reversing the mounting of the tapes 27, 28, and 29. Further, since it is weak to support the pole 32 by cantilevering, it is assumed that the upper end (tip) of the pole 32 is fitted into the annular protrusion 31 at the top of the cover 30 and the tip of the pole 32 is supported.
In addition, although it is the same shape, as shown in FIG. 12, a pair of things are combined so that a slit 38 through which the tape passes is formed through the tape base 35a in the central portion of the support 35, and the tape is there. By reversing 27, 28, and 29 into an S-shape, both the back and front of the tape can be used simultaneously on the front and back surfaces of the tape base 35a of the support 35 without having to bother directly reversing the tape itself. It becomes possible. Further, although there is a gap through which the tape passes when the cover 30 is put on the tape base 35a, if the tape is obstructed and the cover 30 is difficult to cover, a part of the base is cut and the tapes 27, 28, 29 are cut. I don't want to get in the way when the cover 30 is put on. Furthermore, as a point to consider, it is desirable to devise an idea that the pole 32 is not thinned by the rotation of the reels 26 that overlap each other. The completed form is shown in FIGS. In FIG. 13A, A + B + C = D + E + F = G + H + J = 2613-3628A + D + G = B + E + H = C + F + J = 2613-2628, A + E + J = C + E + G = 2613-2628, and the total number (the sum) is the number used and the empty number It will change depending on what you do. The total here does not take into account empty numbers. In addition, the definite sum of the odd order changes for each surface, and thus has a certain range. Even order is constant.
In FIG. 13B, A + B + C + D = E + F + G + H = 3194
I + J + K + L = M + N + O + P = A + E + I + M = B + F + J + N = 3194
C + G + K + O = D + H + L + P = A + F + K + P = D + G + J + M = 3194
In FIG. 14, the 3rd magic square, (3 × 3) 6 plane, 4th magic square, (4 × 4) 6 plane, 5th magic square, (5 × 5) 6 plane, 6th magic square, (6 × 6) Numbers used for the 6th, 8th magic square, (8 × 8) 6th, 10th magic square, (10 × 10) 6 and their definite sum. Again, empty numbers are not taken into account.
In FIG. 15, the fifth magic square, (5 × 5) 6 plane, 7th magic square, (7 × 7) 6 plane, 9th magic square, (9 × 9) 6 plane, 11th magic square, (11 × 11) Numbers used on the 6th, 13th magic square, and (13 × 13) 6 and the definite sum. Again, empty numbers are not taken into account.

The solid magic square polyhedron according to the fourth invention is of a pyramid type and will be described with reference to FIGS. The feature of this toy is that the magic squares are piled up one by one in a pyramid shape in the style of building blocks. As can be seen from FIG. 16, each block 40 that is a building block is a cube or a rectangular parallelepiped, and a step 42 is provided in the support 41 so that the blocks 40 do not collapse easily, and the blocks 40 are sandwiched between the blocks 40. It has become. Furthermore, a certain level of accuracy is required so that the blocks 40 do not rattle. The number entry 43 in FIG. 17 represents the number rotated 180 degrees on the side surface so that it may be stacked in two steps, and the upper or lower surface has a diagonal line so that it may be placed in the corner. It represents along. The same number is represented on each surface of one block. Specifically, as shown in the enlarged portion of FIG. 17, “123” is written on the upper surface along the diagonal line 43a, and “123” is written on the side surface. The numbers are rotated 180 degrees and entered in the upper and lower two columns. Although it can be made numerically whether it is a parent-child magic square or not, it is possible to select a different number. This is also a magic square with a completely different arrangement depending on the choice of the person to be stacked. The foundation is provided with steps of 4 × 4 at the bottom. In the figure, the selection of the numbers used temporarily is N = 1 to 216, but these numbers can be changed freely. Here, A to D layers 1 to 100, E layers 153 to 216, F layers 117 to 152, and G layers 101 to 116 are used. The number of pyramids can be increased to 2 or 3 without being restricted by this number. If the support 41 is changed in size, a 12 × 12 magic square is used outside this number up to the number N = 217 to 360. It is easy to cover it. At that time, the outer definite sum is 3462. In the description of the first to third inventions, it is only necessary to search for a magic square that has already been completed, but this is aimed at completion from the beginning, so it is quite a challenge for those who do not know how to make a magic square. It seems that there is no reference book, so I would like to start with a simple one and enjoy the process of completing it. By the way, the parent and child magic squares are the magic squares when viewed in 4x4, 6x6, 8x8, 10x10. FIG. 18 is a magic square when the pyramid shape of FIG. 17 is viewed from above. Although it is possible to create a pyramid type even if it is not a parent-child magic square, it is more difficult for those who are not accustomed to the parent-child type, so this is the choice. 19 to 21 show the completed form. In addition, although described here as an even-numbered pyramid parent-child magic square, an odd-order pyramid parent-child magic square can also be produced. The basic structure of the even type and the odd type is exactly the same except that the definite sum is different. The even type is currently completed as a pyramid type of up to 18x18, but it is basically possible to create an even type and an odd type regardless of how large the numbers are. Only definite sum is shown. Although the numbers used by the parent and child magic squares hidden in FIG. 20 are shown, some definite sums are not shown, but the results are almost the same, so they are omitted here. If even the numbers hidden inside are included, N = 1 to 454 is used. As a proposer, although it is a toy, it creates a large pyramid magic square that allows people to go inside, and if there is an object that can freely climb in and out, you can feel a different space that entered the magic square I also think that.

A three-dimensional magic square polyhedral ball connection type according to the fifth invention will be described with reference to FIGS. As shown in FIG. 22, the sphere 45 with the numeral 48 is assembled into a three-dimensional shape using the connecting pin 46 in accordance with the order. The number of spheres is 27 for the third-order solid, and the number of spheres 45 is 64 for the fourth-order and 125 for the fifth-order. The spherical body 45 has six tapered holes 47 machined so that it can be connected to the top, bottom, left, right, and front and back. What is different from the content of the polyhedron described in the first aspect of the invention is that the sum (definite sum) at the solid diagonal is a problem. Regardless of the diagonal lines in each plane (the total number of diagonal lines), the total of the solid diagonal lines (including the top and bottom directions) when three-dimensionally superposed from the order NO. Is required to be the same. This completed form is shown in FIG. At this time, the diagonal sums on the plane viewed from only one surface do not necessarily match. It is ideal to agree, but so far it seems that it has not been requested so far. Ordinarily, it is not possible to confirm the numbers that have entered inside directly by simply stacking numbers in blocks. Therefore, in the present invention, it is actually desired to be able to follow not only on paper but also by human eyes. In FIG. 23, NO.1 is a view of FIG. 22 as viewed from above, and NO.2 is a second step as viewed from the top of FIG. NO.3 is the third level when FIG. 22 is viewed from above. NO.4 is the fourth level when FIG. 22 is viewed from above. NO. 5 indicates the lowest level when FIG. 22 is viewed from above. In the case of a fifth order solid, if the number of spheres 45 is used only on the plane, it is possible to make 121 spheres 45 up to the eleventh order plane magic square. If there is a box partitioned without using the connecting pin 46, it is possible to challenge the magic square on the plane with the ball 45 alone. The three-dimensional magic square here is different from conventional polyhedral magic squares, and the order of the vertical, horizontal, and height is determined, and the sum of the solid diagonals is a problem, so it can take other forms. The feature is that it cannot. However, it is possible to take the form of the station number while maintaining the shape as it is while maintaining the sum of the solid diagonal lines. The 3rd and 4th solids are connected like a chain while increasing the number of consecutive numbers. Besides, by making the numbers written on the sphere correspond to Braille, the numbers can be understood by touching them, and it can be a toy for the visually impaired. This completed drawing is shown in FIG.

The solid magic square polyhedron composite diagonal type according to the sixth invention will be described with reference to FIGS. This is also related to Invention 5, but as shown in FIG. 25 (A), 27 sets of solid diagonal magic squares with 3 × 3 × 3 types and 27 definite sums are made separately. To do. The fifth invention simply tries to make 5 × 5 × 5 from the beginning, but here it is made in 27 series, and it tries to make it into a composite 9 × 9 × 9 solid diagonal type . Numbers to be used are from N = 1 to 729, but there are 27 types of definite sums from 1056 to 1134 in three jumps. Obviously, when viewed from a 3 × 3 × 3 magic square, it is of course a solid diagonal type, and even when viewed as a whole 9 × 9 × 9, it is formed as a solid diagonal type magic square. The definite sum of 9 numbers at that time is 3285. In the solid diagonal type, the vertical, horizontal, and height of each surface (9 in 3 × 3 × 3) are definite, and the solid diagonal (4 in 3 × 3 × 3) is definite. . Even if it is a plane type, a synthetic magic square already exists, but it is a solid version. Since it seems natural to make it from the beginning, the 27 magic squares in the shape of Fig. 24 (A) have been assembled from the beginning and numbers have been entered, and they are color-coded. Therefore, the player can play only 27 blocks with the stacking sum being 3285 while considering the direction. There are 648 combinations because there is a direction even if one block is placed on the support base 75. The shape of the block may be the same as in the invention 5, but if possible, the material of the cushion is also considered. Instead of throwing the ball, you can throw them at each other and handle them a little roughly. It is also possible to create a large material by connecting the materials firmly and thinking about the material so that the children can go through like the old jungle gym. I think that this magic square can be made in the same way regardless of the 4th or 5th order. The shape currently considered is that nine sets of magic square blocks 76 are arranged on a support base 75 to form a lower stage. Next, the same thing as the support stand 75 is put on the magic square block 76 arranged in the first stage to prepare for making the second stage. Further, nine sets of magic square blocks 76 are placed on the second stage, and the same thing as the support base 75 is similarly placed on the top to prepare for the third stage. When the remaining 9 sets of the third magic square block 76 are loaded and the sum is 3285, it is completed. The shape of the support base 75 is made of transparent plastic, and 64 holes 77 corresponding to the interval of the spheres of the block 76 are formed in a number of 8 × 8. The aim of 64 instead of 81 is that it is the smallest material and no extra parts are visible. Accordingly, the outermost sphere 76 does not have a base part and floats in the air. By using the same type of support base 75 for all parts, the parts can be shared. Although I want to make it with cushions, the transparency of the support stand 75 is still an important factor. In addition, the support base 75 is hollow and has 64 holes at the same interval on both sides, so that the upper and lower spheres 76 are firmly supported. Furthermore, although not confirmed yet, this 9 × 9 × 9 solid composition diagonal block is seen as one lump, and 27 sets are made once again in the same way as a 27 × 27 × 27 solid double If you make a synthetic magic square and see it, you're predicting that you can make a larger synthetic three-dimensional magic square. The numbers used at that time are N = 1 to 19683. This method is easy for large orders because it is possible without using special software, although there is a restriction of order multiple. By the way, as shown in FIG. 26 (A), 4 × 4 × 4 is multiplied by 4 and the diagonal magic square of 16 × 16 × 16 uses numbers from N = 1 to 4096, and its definite sum is 32776. . In FIG. 26B, if 5 × 5 × 5, the number of diagonal magic squares used by multiplying by 5 × 25 × 25 × 25 is N = 1 to 15625, and the definite sum is 195325. If you replace the numbers with weight distribution, you can make a very balanced cube with the center of gravity in the center. This is shown in FIG. Since it is a cube, there are six ways to place it without any face up. Of course, any number of ways of thinking and how to make the order is possible, so further enlargements are possible, and it is common to all inventions, but here even numbers can be made only of odd numbers. In addition, the solid diagonal type basically does not matter the sum (definite sum) of the diagonal lines in the plane, but since the sum of all the solid diagonal types on the plane diagonal does not necessarily coincide, It must be understood that there are some matches including corners.

The three-dimensional magic square polyhedral circle type according to the seventh invention will be described with reference to FIGS. As shown in FIG. 27, a ring 56 with magic squares wound around the support 55 in the shape of a sphere is wound by the order, and is mounted so as to freely rotate around the support 55. ing. A numbered frame 58 is embedded in the ring 56, and the total number (definite sum) of the frames 58 on the ring is the same in all the rings 56. The ring 56 is fixed by a set screw 57 attached at the top and bottom, so that the ring 56 can rotate but cannot be removed. In this circle, when looking only at the upper and lower hemispheres, the sum of the numbers arranged on the ring 56 (the sum) and the sum of the numbers arranged in the north-south direction (the sum) seem to coincide simultaneously by turning the ring 56. It has become. When viewed in a sphere, the sum of the upper half and the sum of the lower half also coincide vertically. Even when the number of spheres is increased, it is consistent with the definite sum of each hemisphere. Depending on the type of circle, the number of frames 58 on the ring 56 may be different from the number of frames 58 in the north-south direction, but the number of frames 58 on the ring 56 and the number of frames 58 in the north-south direction are still The two are in harmony with each other. As shown in FIG. 28, eight rings are attached to one sphere with eight numbers in the east and west directions. Therefore, the number used is N = 1 to 192 because the sphere is triple. The definite sum at that time is 772. The total number of numbers in one sphere is 6176 for all three, and 3088 for a hemisphere. In FIG. 29, ten rings are attached, and the number of used balls is N = 1 to 400, and the definite sum is 2005 because four balls are used. Therefore, the sum of the numbers in the hemisphere is 50, which is 100250.
As for the shape of the sphere, it is possible to simply make holes in the east-west direction and north-south direction by the order of the sphere, and insert the number 58 with the numbers there, but it is simple, but for the first person Since it is likely to be a little difficult result, FIG. 27 shows an attempt to search only the sum of the numbers in the north-south direction by rounding in the north-south direction. Of course, it is possible to divide the inside of the support body 55 to make it hollow, and it is possible to put a favorite thing such as confectionery in the inside. Also, like a three-dimensional jigsaw puzzle, a piece 59 can be fitted with a magnet, and if a three-dimensional picture (such as a globe) is completed, the circle is automatically completed. Furthermore, it is the same round slice, but when the head of the doll or favorite person is cut round and the head is in the correct position, it is loose as a circle, but when it is completed as a circle, the person's head changes. It is also possible to think of a way to end up in a different shape.

The solid magic square polyhedron according to the eighth invention is of a block type and will be described with reference to FIG. In the present invention, a cube 61 for supporting the magic square piece 62 on the pedestal 60 covers the pedestal 60 and is hollow. Characteristically, magic squares are made on the side and upper surface of the cube 61 excluding the pedestal 60, but the piece 62 on which this number is written has magnetism, and the iron coated on the surface of the cube 61. The piece 62 is attached to the side surface of the cube 61 by using powder or a thin iron plate. The difference from the first invention is that although it is a polyhedron, empty numbers are not used, and it is normal to think from the beginning of odd-numbered and even-order magic squares. It's a little harder, but it's also a form of thinking about the original magic square, so I'd like you to complete the entire polyhedron, not just one side. The idea here is that it is a magic square toy and can also be used as promotional goods. The surface of the ink containing iron powder applied to the surface of the cube 61, or a thin iron plate, This is a technique of having a landscape photograph or a face photograph of someone and pasting a magic square piece 62 thereon. The four sides are used for advertising purposes and printed and used as desired. The pieces 62 are arranged and arranged as magic squares on the four sides, and are attached with magnetic force. In other words, it is a polyhedral magic square made of four or five faces. As shown in FIG. 30, this cube is made by simply folding a sheet of paper, plastic, or metal 61 and can be printed on the surface. In some cases, the inside of the box can be used for advertising. It is possible to increase the amount of information if the piece 62 is color-coded and put some information on it, and it is possible to increase the amount of information and it can be easily discarded by becoming a toy, and even individual parts The advantage is that the shape is simple and inexpensive. Applying magnetic powder on cardboard and printing on it, and then attaching the magic square with a magnet, if it is impossible in terms of magnetic force and strength, it is possible to print on a thin iron plate as much as possible. On the contrary, it seems easy to bend and mold it into a box shape, but if you don't like folding, it is an idea to use that part as a hinge. The number is 64 for a 4th order tetrahedron and 80 for a 5th face. The inside is hollow, so anything can be put in it and if it is made up, it can be used as a container. Some people may be concerned that the first invention may have the same numerical arrangement, so if you explain, the first to third inventions basically consider hexahedrons and more. In the invention of No. 8, it is limited to a tetrahedron or a pentahedron. Therefore, the content will not be the same as long as the number of faces is different. In the case of the fourth order, there is a possibility that the number of the first side is the same as the method of making N = 1 to 16 and the second side, N = 17 to 32, one by one, but here the method is Will not take. In the even-order system, there is a problem that the result is limited because the definite sum of all faces is the same, but there is a way to change the sum of little by little like the odd-order system It's not impossible. Since that place is determined by the sense of the creator, it cannot be determined here. FIG. 32 shows a magic square decomposition of numbers 1 to 3 in the form of fractions as a completed form. In (A), the sum of 45 numbers is 1, 1 in (B), and 2 in (C). As shown at the left end in (A), A + B + C = aa + bb + cc = Ka + Ki + Ku = A + aa + Ka =
B + bb + Ki = C + cc + Ku = A + bb + Ku = C + bb + Ka = definite sum. The definite sum of each of the 2nd to 5th surfaces is shown below. In FIG. 33, the 3rd to 5th magic squares in 5 planes are shown as 1 to 9 in total. Of course, it is free to set the number of faces, order and total number. It ’s possible that something with that style is better here. This magic square decomposition is also possible with a solid diagonal type as shown in FIG. 24, and any number can take the form of a fraction in the same way with a solid type without being particular about the plane. Here, the magic square decomposition is performed from 1 to 9 in total, but it is also possible to decompose the definite sum from 1 to 9 in the even number. In addition, it is recommended that you try out one of the methods, because it is possible to use something different from the usual array using quaternary or quinary, depending on the order. In general, how to give hints and issues is a problem in all inventions, but I would like to collect consensus and examine it. As shown in FIG. 31, there is a demerit in that it is inconvenient to repeatedly use the metal by bending it into a box and making it bulky. I also considered a form in which the piece 62 is made of aluminum without using magnetic force, and a hook 63 is provided by bending a part and inserted into the small slit 64 of the cube 61. The cubic box 61 is normally in the form of a sheet 65 and is kept in the shape of the cube 61 by being inserted into the groove 66 of the pedestal 60. It can be used as a container and can be used as a four-sided photo stand by adding a rotation function 67 to the pedestal while it is not used as a magic square. In this case, the photo cover 68 is also necessary, but at the same time, the base 60 is made a little trapezoid 69 to make the photograph easy to see, and the cube 61 is also made a little trapezoid 70. Overall, this shape can also be a simple hexahedron box, so it can be a puzzle-shaped 3D magic square polyhedron with just a few printed numbers as a completed magic square box just printed. . I think this is meaningful because the degree of perfection can be changed by the amount that the number is removed. Although it is a world where if you just give an answer because it is the present era, you can do it as it is if you use a PC, I think it will be an opportunity to think about the process of how old people struggled.

The solid magic square polyhedron according to the ninth aspect of the invention is of the even pyramid free exchange parent-child type and will be described with reference to FIGS. This is similar in shape to the pyramid shape described in the fourth invention, but the number arrangement is completely different. Since the structure is exactly the same, explanations on this point are omitted, but the reason for the difference is in how to make the array. In the fourth aspect of the invention, a single pyramid is formed by using a plurality of surfaces of 8 × 8, 6 × 6, and 4 × 4, respectively, but the pyramid is formed by using consecutive numbers for each surface. It is not possible to exchange orders of the same order. Although this is a conventional way of creating a normal parent-child magic square, in the ninth invention of this time, the frames of each degree can be freely exchanged including the direction in the pyramid for each frame. ing. This is possible because definite sum is the same total in each same order frame, but if you express it differently, even if you try to overlap 12x12 using numbers from 433 on this work Even if you can make a parent-child magic square, it will not be compatible with the parent-child magic square in the lower row of 10x10. It is possible to make it as an interchangeable type in the newly overlapped part, but the consistency from the beginning is lost. However, it is not without a method, and if the number to be used is expanded not only to a positive integer but also to a negative integer, it is possible to increase it while maintaining consistency as an exchange type. In other words, if you decide the size to make from the beginning and choose the number to use, you can make it only with a positive number and make it an exchange type, but if you try to make an exchange type by adding more numbers, it will be a negative number Therefore, a plus and minus number are needed at the same time. This time, the twin was made only for the purpose of simply increasing the number of compatibility, and there is no need for a twin. The numbers used are 1 to 432, but 10 × 10 is 2 frames and the definite sum is 2165 (A and B). 8x8 has 4 frames and its definite sum is 1732 (Figures C to F). 6 × 6 has 6 frames and its definite sum is 1299 (G to M diagrams). 4 × 4 has 8 frames and its definite sum is 866 (N to V diagrams). Within this range, the interchangeable parent-child magic square can be established without any problem even if the frame is exchanged freely. It is natural that it is possible to make a larger interchangeable model by changing the shape of the base part, and there is no restriction on the order and the number of faces. Although not shown in the figure, if 4 × 4 is 6 faces, 6 × 6 frame is 2 faces, and the number to be used is N = 65 to 200, the 4 × 4 definite sum is 530, 6 × 6 definite sum Becomes a 795 interchangeable type. Furthermore, if one side of 8 × 8 frame is added and N = 65 to 228, the definite sum of 4 × 4 is 586, the definite sum of 6 × 6 frame is 879, and the definite sum of 8 × 8 is 1172. It becomes an interchangeable parent-child magic square. This type is unavoidable that it takes some time to produce because it needs to be recreated from the beginning each time the number used changes. By the way, if this interchangeable mold is made small and light, it will be an earring or pendant that is likely to be liked by mathematics girls using the four sides.
Next, although not shown in the figure, 48 6 × 6 exchange molds are made, and 8 cubes using 6 of them are made into a composite 12 × 12 × 12 free exchange mold. It is also possible to create a solid magic square that can be freely combined. The definite sum of 12 numbers at that time is 10374. Although it seems necessary to keep the numbers upward, it seems worth considering.
The next is a free-exchange parent-child type with an odd-order frame, which will be described with reference to FIGS. This is not exactly the same as an even number and the neck is in the 3x3 part. It is the same as the even type that you can make multiple exchange parent and child types for surfaces with 5x5 frames or more, but for 3x3 you can make only one surface and create multiple surfaces with the same definite sum. It doesn't come true for now. I think this is due to the difference between the odd and even natures, but the work I put on this time uses N = 1 to 153, and the 9 × 9 frame is 2 frames, and the definite sum is 693 (Figures A and B). The 7 × 7 frame is 2 frames, and the definite sum is 539 (C and D diagrams). 5 × 5 is 2 frames and the definite sum is 385 (E and F diagrams). 3 × 3 is one frame and the definite sum is 231 (FIG. G). Combining any of these completes an interchangeable odd-parent and child magic square with a sum of 693. Although the number of faces is still not enough, if this is applied to the pyramid shape, except for the 3x3 part which becomes the top surface, the other 3x3 can not be made and it will be hollow, so you can put the box there Consider mummy, white bone, or elaborately modeled human body. Or it can be a flower, another figure, or a doll. It seems that it is difficult for inexperienced people to make these forms from the beginning because there are no examples made in the past, but examples are written from the beginning. The other half should be considered with reference to the correct answer example, and it may be a method to disassemble and understand the structure again.

The solid magic square polyhedron according to the tenth aspect of the invention is also related to the first aspect of the invention, and will be described with reference to FIG. The structure shown in the invention 1 is a little complicated, and therefore, the inventions 2 and 3 are considered. However, the belts 2, 3, and 4 used in the invention 1 are used as they are, and the support body in the shape of a wheel as a whole is used. FIG. 39 shows that the structure can be easily divided. Here, the support body 71 is divided into six parts, and the belt stoppers 72 divided into six parts are simply fixed to the trapezoidal support bodies 71 with screws 73 alternately. That is, adjacent support members 71 divided into six parts are connected one after another by belt stoppers 72 divided into six parts. The belt stopper 72 forms a flange at the upper and lower ends. This makes it possible to handle large orders without changing the length of the belt.

DESCRIPTION OF SYMBOLS 1 Support body 2 Belt 5 Inner ring 6 Upper lid 7 Lower lid 14 Cover 15 Support body 16 Tape 20 Reel 25 Support body 26 Reel 27 Tape 30 Cover 35 Support body 36 Reel 40 Block 41 Support body 42 Step 45 Sphere 46 Connection pin 55 Support body 56 ring 58 frame 59 piece 60 pedestal 61 cube 62 piece 68 cover 71 support 72 belt stop

Claims (7)

  It is a solid magic square that forms magic squares on each side of multiple faces of a three-dimensional shape, and if the arrangement of numbers on one side is changed, the arrangement of numbers on the other side is also changed in conjunction with it. Three-dimensional magic square characterized by being.   The three-dimensional magic square according to claim 1, wherein a belt or a tape in which a plurality of numbers are arranged in a length direction is attached so as to be able to rotate over a plurality of surfaces.   A three-dimensional magic square that forms a magic square on at least one surface of a three-dimensional shape, and is a solid having a belt or tape on which a plurality of numbers are arranged in the length direction so that the belt or tape can be rotated over a plurality of stages. Magic square.   It is a three-dimensional magic square that forms a magic square for each of the whole and the interior, and is configured to form a magic square by stacking cube pieces, which are stacked to form a plurality of steps. A solid magic square constructed in a pyramid shape so that the area on the upper side becomes narrower.   It is a 3D magic square that forms magic squares on the outside and inside of the 3D, and it is configured to form magic squares by placing the tops in a 3D, and the tops are arranged with a space between them. A solid magic square that is configured so that the numbers of the top internal arrangement can be seen, and that the numerical arrangement on the solid diagonal is also a requirement for magic square formation.   For the outer and inner surfaces of the solid, 27 of the numbers from 1 to 729 are combined to form a cube-shaped solid magic square, and 27 sets of blocks with the same total in the solid diagonal are made. All the numbers in the block are filled in, but they are configured as one piece, and the whole combination of them is also a cube-shaped solid magic square, depending on how the block magic square is arranged. A three-dimensional magic square characterized by the fact that magic squares can or cannot be formed.   A three-dimensional magic square in which a magic square is formed on each face of a plurality of faces in a three-dimensional shape, and numeric display pieces can be detachably arranged on each face.

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