Four Color Theorem Board Game

Abstract

To prove a four-color theorem, and to provide a board game that allows one to enjoy the game by thinking and competing using a uniform rule utilizing the means and a rule of the four-color theorem.SOLUTION: In a board game based on a four-color theorem comprising multiple board surfaces having a map consisting of multiple grids and multiple pieces having four knobs having different colors disposed based on a rule of a four-color theorem, teams are divided into an attacking side team and a defensing side team, and they compete each other for the total number of pieces placed on the board surfaces.SELECTED DRAWING: Figure 16

Description

[Technical Field]

[0001] The present invention relates to a toy in which a plurality of pieces are arranged according to the rules of the four color theorem on a board on which a grid map is drawn. [Background technology]

[0002] There have been many board games to date, but the four color theorem, a difficult mathematical problem, has not yet been proven using a method that does not require a computer. By simultaneously disclosing this proof with this application, we will provide an original board game that can be played according to the rules of the four color theorem. DISCLOSURE OF THE INVENTION [Problem to be solved by the invention]

[0003] The four color theorem requires that all areas of the entire map be colored in four colors, so the method of achieving this was itself a challenge. However, the problem and the coloring process itself were not difficult, so although it was not possible to come up with a complete answer, the process was enjoyable, like solving a quiz.

[0004] The present invention aims to solve the above problems. Specifically, the present application aims to prove the four color theorem and provide a board game that allows players to enjoy thinking and competing based on unified rules that utilize the means and rules of the four color theorem. [Means for solving the problem]

[0005] The present invention provides a board game of the four color theorem, which is made up of a board with a map consisting of a plurality of squares and a plurality of pieces with four knobs of different colors arranged according to the rules of the four color theorem, by proving the four color theorem by a means showing how to arrange the four colors over the entire determined area, rather than by using a computer to check the number of all cases of the four color theorem one by one, and by providing a board game of the four color theorem with rules based on that means. [Effects of the Invention]

[0006] The board game of the present invention has the advantage of solving the four color theorem in a previously unproven way that does not use a computer, and of providing an unprecedented, original and enjoyable experience. BEST MODE FOR CARRYING OUT THE INVENTION

[0007] From now on, I will reveal and explain the details of the four color theorem. First, when a plane is divided around a center point by drawing multiple lines from the center point toward the circumference of a circle, these lines divide the plane into regions, and two adjacent regions are called interference regions. The range of these lines is called an interference line. The maximum number of regions that all interfere with each other through interference lines that share a point is three. Figures 1 to 3 show a comparison example, and are simplified diagrams showing the joints of regions that share a point and interference lines. If we define the vicinity of the interference line, shown in Figure 1, where regions interfere with each other in a Y-shape due to the division by interference lines that share a point as Y-shaped interference, and define the case in Figure 2 where all regions share only a point up to infinity as V-shaped interference, then the division of the plane is limited to V-shaped interference or Y-shaped interference between three regions. However, if we consider the area outside the overall view, which is the range of the designated area where the regions extend side by side, as a single region, the division of the region is also Y-shaped. However, as shown in the simplified diagram in Figure 3, this is defined as T-shaped interference between two regions to distinguish it from Y-shaped interference. Cases like Figure 5 are represented more clearly by replacing them with Figure 3, focusing on the joints of regions that share a point. Furthermore, as shown in Figure 6, even if we treat the circumference as a combination of extensions of multiple boundary lines drawn from the center point of the circle toward the circumference without changing the division method and then try to freely deform these extensions, the extensions of multiple boundary lines will not intersect with each other, so there are only two ways to form an area: either the extensions of adjacent boundary lines will connect with the order in which they are lined up along the circumference without changing where they are joined, or the extensions of the boundary lines will connect with the boundary lines of the next area lined up along the circumference so as to surround several other areas in a U-shape, and form an area.In either case, the principles of dividing areas on a plane are not related to the overall shape or size of each individual area. In other words, all areas in the method of dividing a plane share some interference lines and points, and the combination of areas is limited to V-shaped adjacent or Y-shaped interference and T-shaped interference, which are not all considered to be interfering areas. From the above, if we consider any Y-shaped or T-shaped interference region as a determined element, and the number of regions and the region arrangement pattern outside of that as uncertain elements, then the maximum number of regions that all interfere with each other through interference lines that share a point, as shown in Figure 1, is 3 + uncertain elements. By the way, as shown in Figures 4 and 9 when the division of the plane is one region, by including several regions within a region within a deterministic element, it is possible to show an example where the maximum number of regions where all of them interfere with each other is 4. Also, Figures 7 and 8 are of a similar form, and can show an example where the maximum number of regions where all of them interfere with each other is 4 simply by being surrounded by multiple regions. Therefore, similarly, by surrounding some of the uncertain elements with a region so that they interfere with the region of the deterministic element, it is possible to turn the uncertain element into a deterministic element, and it is possible to determine that the maximum number of regions where all of them interfere with each other is 4. This is because these examples show all the ways in which the maximum number of regions where all of them interfere with each other is 4, and since the interference with the uncertain elements can be considered as just one region in Figures 4 and 9, and Figure 7 can be replaced with Figure 1 and Figure 8 with Figure 9, it is possible to determine that all of them interfere with each other. This is because the number of interfering regions is limited to a maximum of four combinations, and they do not contribute to increasing the maximum number of mutually interfering regions or the uncertainty factor. On the other hand, the combination of regions that all interfere with each other through interference lines sharing a point is limited to T-shaped interference and three-region Y-shaped interference, so as shown in Figure 10, one region in an uncertain element that interferes in a Y-shaped manner along two regions in a deterministic element can be established as a new third region in the deterministic element and absorbed. Similarly, as shown in Figure 11, two regions in a deterministic element and one region that are adjacent to each other through Y-shaped interference or V-shaped adjacency can be repeatedly established as new deterministic element regions, so that the Y-shaped interference and V-adjacent regions are linked together like numbers, and can further be formed into a band-like layer as shown in Figure 12. References 14 and 17 in Figure 12 show T-shaped interference. Therefore, by targeting three regions of deterministic elements and overlapping strip-like region layers along this central target as shown in Figure 12, all specified regions can be turned into deterministic elements. Note that it is also possible to expand the region of deterministic elements radially, but here we will limit ourselves to the method of creating strip-like layers so that the region is continuously aligned with the deterministic elements. The numbers in the figures shown in FIGS. 1 to 12 are numbers assigned to different colors. To divide this into four colors, you can freely decide how to arrange the four colors in the area, so you can do so according to the following rules. First, to avoid the same color number due to area interference, four is the maximum number of areas where all areas interfere with each other, so if the areas are arranged in advance so that two are facing each other vertically and horizontally, as in Figure 2, even if the vertical interference line shifts to the dotted line position, two colors can be arranged in a regular alternating band shape, as in Figure 13. Next, if one area is determined one by one from the uncertain element along the two areas in the determined element as a new third area of ​​the determined element, even if three of the four colors are assigned, there will always be one extra color number that is in flux, so even if the number color of one area that interferes with three areas where three different colors have been determined in advance is decided later, as in Figure 11, the remaining color number can be arranged in an irregular manner. Therefore, although there is not one order in which all the regions are made into determinate elements, as shown in Figure 12, if you arrange the regions in two colors alternately and regularly in the circumferential direction for each band-like layer centered on the target region, and then decide and arrange the regions one by one in the order in which they were determined as determinate elements, you will be able to fill all the regions with only the four number colors, avoiding the same color numbers in the interference regions, as shown in Figure 12. The proof of the four color theorem requires that any area always has at least one area of ​​a different color between it and other areas of the same color, and that all four areas of different numbered colors up to the entire specified area can be arranged in order without interfering with areas of the same numbered color.For example, if we decompose the three areas 1, 2, and 3 in Figure 1 by giving them colors 1, 2, and 3 respectively, and then aligning colors 2 and 3 as the determining elements around color 1 for any area, we can determine the numbered colors of the areas one after another in the order of Figure 12, as shown in Figure 14, and show how to arrange them in colors 1 to 4. In other words, a strip-like layer of all color 2 and color 3 areas along any color 1 area, and a strip-like layer of color 1 and color 4 areas along that strip-like layer are determined, and the entire specified area constructed by repeatedly overlapping these layers alternately is a method in which colors 1 to 4 are arranged in order, and Figures 13 and 14 show that the four color theorem holds no matter how many layers are overlapped. Therefore, the four color theorem is proven to be valid. Finally, Figure 15 shows an example in which it is impossible to use a computer to check the number of all cases of the four color theorem one by one. This is because this method is a way of showing how to arrange the four colors in the entire determined area, whereas the method using a computer is a way of assuming and investigating the number of all cases for the combinations of an infinite number of undetermined areas, including the factor of the infinite number of interference lines. Next, an embodiment of the present invention will be described with reference to the drawings. The Four Color Theorem board game of the present invention is a board game consisting of a board with a map consisting of multiple areas and multiple pieces with four picks of different colors that are arranged according to the rules of the Four Color Theorem. The players are divided into an attacking team and a blocking team and are placed on the board. It is a game in which players compete to see who can get the most pieces. In the first example, following the rules of the four color theorem, the attacking team starts in any territory, followed by the blocking team in any territory. The attacking and blocking teams alternate placing pieces one per territory. The rules for placing pieces are as follows: the attacking team can only place pieces in territories adjacent to territories already occupied by their own team, while the blocking team can place pieces in territories adjacent to territories occupied by both teams. In accordance with the four color theorem, pieces must be selected from the four colors so that they do not overlap with other adjacent territories. Pieces are placed in territories by inserting the effective pick color shown in Figure 17 onto the top and the opposite pick color into the hole shown in Figure 16. Since the positions of the holes within the territories are predetermined, multiple pieces can be placed on the board without interfering with each other. The game continues until both teams run out of pieces on the board, alternating between attacking and blocking. The team with the most total pieces on the board wins. Unlike the Four Color Theorem, where you can decide on four colors by yourself, this is a difficult and fun game where you have to cover the entire area with four colors. [Brief explanation of the drawings]

[0008] [Figure 1] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 2] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 3] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 4] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 5] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 6] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 7] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 8] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 9] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 10] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 11] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 12] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 13] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 14] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 15] FIG. 1 is an explanatory diagram of the operation of the Four Color Theorem board game according to the first embodiment of the present invention. [Figure 16] FIG. 1 is a perspective view of the Four Color Theorem board game according to a first embodiment of the present invention. [Figure 17] FIG. 1 is a perspective view of a four color theorem piece according to a first embodiment of the present invention.

Claims

[Claim 1] A board game comprising a board having a map made up of a plurality of areas and a plurality of pieces each having four knobs of different colors, the pieces being arranged on the board according to the rules of the four color theorem, the plurality of areas each having a hole, and each piece having one knob inserted into the hole, the pieces each having the four knobs of different colors at the tip of a cross-shaped rod.

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