Repeating this logic shows that three, then all four triangles with solid corners are connected in one tetriamond. But since each vertex has only five neighboring vertices, then whether or not the two triangles just mentioned share an edge, there must be two triangles that actually do share an edge.
By the Pigeonhole Principle, we can say that at least one vertex belongs to two or more triangles. Note the four tiles have twelve corners total. There are four tiles with no corner holes these must share the six no-hole vertices, and can do so in one of two ways. With 2-sided rotational symmetry, there are 20 ways - can they make an icosahedron with edge matching? Bryce Herdt: "First, each vertex must be surrounded by either 0 or 5 corner holes, and there must be six of each vertex. They fit nicely into a hexagon with matching. Raster Tournament Dice Deadly Rooms of Death Fair Dice Chessboard Tasks A Zillion Connection Games Keen Approximations NUMB3RS TV show Cyclopedia of Puzzles Sliding-block Puzzles Rulers and Gracefulness Hoffman-Singleton Game Ten Trillion Zeta Zeros Evil Numbers Manifolds in Genesis 64K or Less - Demoscene Modern Burr Puzzles Egyptian Fractions Nob Yoshigahara 2D Turing Machines WireWorld Multiplication Crossword Rules The Quantian Distribution Integer Complexity Supermagnetic Polyhedra Gaussian Numbers Number Games Guilloché Patterns Prime Megagap Wolfram Functions Site Cubic Symmetric Graphs Superflu Modeling Domino Graphs Sequence Pictures Square Packing Multi-state Mazes Loculus of Archimedes Matrix Revolutions Möbius Function Paterson's Worms RevisitedĦ-punch Triangles There are 24 ways to put spots on the corners/edges of an equilateral triangle, with 1-sided rotational symmetry. Math Games Social Golfer Problem Wolfram Demonstrations Dodecahedral Tilings Power Sums Melbourne: City of Math Snakes on a Plane Prime-generating Polynomials The Fano Plane G4G7 Kobon Triangles Beautiful References Times Square Magic From Keyfobs to Ringtones Sudoku Variations Vector vs.
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