Here's a neat trick involving all the numbers from 1 to 26, and three of the five Platonic Solids, the most basic polyhedral shapes. Let's start with the tetrahedron. A tetrahedron, sometimes called a triangular pyramid, is a shape made up of four corners, four equilateral triangular faces, and six edges. Since each face is a triangle, it also has a total of 12 angles on its four faces. So the tetrahedron has a total of 26 components, (4 corners + 4 faces + 6 edges + 12 angles = 26). These components can be numbered from 1 to 26 in a special way.

The basis of the trick is to use three pairs of numbers in order to create a value for every part of the tetrahedron. The three pairs are: 1 & 2, 3 & 6, 9 & 18. These six numbers will be placed on the six edges of the tetrahedron. Each edge is always directly opposite another edge; that is, if you draw a line through the center of the object starting from one edge, you will always reach another edge on the opposite side.

Select any edge and place the number 1 on it. On the opposite edge place the number 2. Select any of the remaining edges and place the number 3 on it, and then place the number 6 on the opposite edge. On the last two edges place the numbers 9 and 18. Now you're ready to determine all the rest of the numbers, and where they go on the tetrahedron.

Every corner has three edges that meet at it. Add up the value of the three edges and give that number to the corner. Every side has three edges surrounding it. Add up the value of the three edges and give that number to the face. Every angle on the faces is formed by two edges meeting there. Add the value of these two edges and give that number to the angle.

When you are done, you will discover that you have used all the numbers from 1 to 26 without repeating any numbers! There are two possible tetrahedra that can be made this way. Can you find them both?

This trick can also be done on a cube, but without using the angles of the faces. A cube has six faces, eight corners, and twelve edges; 6 + 8 + 12 = 26. Using the same three pairs of numbers we started with above, we can give values to the six faces. Place the number 1 on a face, and the number 2 on the opposite face. Place the number 3 on any of the remaining faces, and the number 6 on the opposite face. Then place the numbers 9 and 18 on the remaining faces.

Just like before, we will add these six numbers together to create all the others. Every edge of a cube is the meeting place of two faces. Add the numbers of these two faces and give that total to the edge. Every corner of a cube is the meeting place of three faces. Add the numbers of these faces together and give that total to the corner. Do this for all parts of the cube, and you will use all the numbers from 1 to 26 without any repetitions. Unlike the tetrahedron, there is only one way to accomplish this feat on a cube.

The technique will also work on the octahedron, which has six corners, eight sides, and twelve edges; 6 + 8 + 12 = 26. In this case the three pairs of opposite numbers are placed on the six corners of the octahedron. As before, place 1 and 2 on opposite corners, 3 and 6 on opposite corners, and 9 and 18 on the remaining corners. Each of the twelve edges of the octahedron connects two corners. Add these two corners to find the number for that edge. Each of the eight triangular sides of the octahedron is surrounded by three corners. Add those three corners together to get the value for that side. When completed, you will again have used all the numbers from 1 to 26 with no repetitions. As with the cube, there is only one way to accomplish this.

The sharp observer will have noticed that the special set of 3 opposite pairs are used for the edges of a tetrahedron, the faces of a cube, and the corners of an octahedron, which may reveal something important about the relationships of points in space. Also, for the die-hard puzzle solver, the solutions to the above figures may be written in Base 3 numerals, with very interesting results.

The basis of the trick is to use three pairs of numbers in order to create a value for every part of the tetrahedron. The three pairs are: 1 & 2, 3 & 6, 9 & 18. These six numbers will be placed on the six edges of the tetrahedron. Each edge is always directly opposite another edge; that is, if you draw a line through the center of the object starting from one edge, you will always reach another edge on the opposite side.

Select any edge and place the number 1 on it. On the opposite edge place the number 2. Select any of the remaining edges and place the number 3 on it, and then place the number 6 on the opposite edge. On the last two edges place the numbers 9 and 18. Now you're ready to determine all the rest of the numbers, and where they go on the tetrahedron.

Every corner has three edges that meet at it. Add up the value of the three edges and give that number to the corner. Every side has three edges surrounding it. Add up the value of the three edges and give that number to the face. Every angle on the faces is formed by two edges meeting there. Add the value of these two edges and give that number to the angle.

When you are done, you will discover that you have used all the numbers from 1 to 26 without repeating any numbers! There are two possible tetrahedra that can be made this way. Can you find them both?

This trick can also be done on a cube, but without using the angles of the faces. A cube has six faces, eight corners, and twelve edges; 6 + 8 + 12 = 26. Using the same three pairs of numbers we started with above, we can give values to the six faces. Place the number 1 on a face, and the number 2 on the opposite face. Place the number 3 on any of the remaining faces, and the number 6 on the opposite face. Then place the numbers 9 and 18 on the remaining faces.

Just like before, we will add these six numbers together to create all the others. Every edge of a cube is the meeting place of two faces. Add the numbers of these two faces and give that total to the edge. Every corner of a cube is the meeting place of three faces. Add the numbers of these faces together and give that total to the corner. Do this for all parts of the cube, and you will use all the numbers from 1 to 26 without any repetitions. Unlike the tetrahedron, there is only one way to accomplish this feat on a cube.

The technique will also work on the octahedron, which has six corners, eight sides, and twelve edges; 6 + 8 + 12 = 26. In this case the three pairs of opposite numbers are placed on the six corners of the octahedron. As before, place 1 and 2 on opposite corners, 3 and 6 on opposite corners, and 9 and 18 on the remaining corners. Each of the twelve edges of the octahedron connects two corners. Add these two corners to find the number for that edge. Each of the eight triangular sides of the octahedron is surrounded by three corners. Add those three corners together to get the value for that side. When completed, you will again have used all the numbers from 1 to 26 with no repetitions. As with the cube, there is only one way to accomplish this.

The sharp observer will have noticed that the special set of 3 opposite pairs are used for the edges of a tetrahedron, the faces of a cube, and the corners of an octahedron, which may reveal something important about the relationships of points in space. Also, for the die-hard puzzle solver, the solutions to the above figures may be written in Base 3 numerals, with very interesting results.

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