![]() ![]() Where, , and are respectively the number of faces, edges, and vertices the polyhedron has. You get the answer using the Euler characteristic. If you start creating links, and keep adding links until no more links are possible, then regardless of how you decide which links to make, you always get the same number of links, and the same number of control fields. So a question you might ask is, for some collection of portals, is there an optimal way to link them? Can you make more links or more control fields if you do it one way than if you do it another?Īnswer: No. Links are line segments, control fields are Euclidean plane triangles.) But for our purposes we pretend we’re working in a plane. (Technically, if the Earth is regarded as a sphere, presumably links are great circle segments and control fields are spherical triangles. ![]() If you want to link two portals, and a straight line segment between them intersects some other link, then tough luck: you can’t link them. One rule of linking, though, is that links can’t cross. So if you’re trying to level up, one way is to find a lot of portals close together and circulate between them, capturing them and linking them. If three portals are mutually linked with three links, forming a triangle, then they make a control field, and that’s a good thing if only because you get a lot of points for doing that. In the game Ingress you capture portals (by visiting real world locations), and then you can link pairs of portals together. In each case what’s the best strategy for clearing the board? Can you clear the board, and in how few moves? For bonus points, can you find ways of winning with the last peg left in a given position? Continue reading “Pegs jumping” → There are ten pegs but really only three pegs to consider removing: A corner peg like 1, an edge peg like 2, or the center peg, 5. Keep going until you’re left with only one peg, or there are no legal jumps. That’s a move, but if you can jump two or more pegs in succession with the same peg (removing them all) that counts as a single move too. For instance if peg 1 is removed then you can jump peg 4 over peg 2 into position 1, or peg 6 over peg 3 into position 1. Now start jumping pegs, checkers style, but in any of six directions. Start with ten pegs in ten holes arranged in a triangle: Matt Parker’s newest Maths Puzzle asks you to provide the shortest solution to a triangular peg jumping puzzle. ![]()
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