Lesson A bug lives on a corner of a cube
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<H2>A bug lives on a corner of a cube</H2> <H3>Problem 1</H3>A bug lives on a corner of a cube and is allowed to travel only on the edges of the cube. In how many ways can the bug visit each of the other seven corners once and only once, returning to its home corner only at the end of the trip? <B>Solution</B> <pre> A unified geometric-combinatorial description can be created for all possible configurations. {{{drawing(300,300,-1,4,-1,4, rectangle(0,0,2,2),triangle(2,2,3,3,2.5,2.5), green(rectangle(1,1,3,3)),triangle(0,2,1,3,0.5,2.5), triangle(1,3,3,3,2,3),triangle(2,0,3,1,2.5,0.5), triangle(3,1,3,3,3,2),green(triangle(0,0,1,1,0.5,0.5)), circle(0,0,0.05),locate(-0.2,0.2,1),locate(-0.2,2.1,2), locate(0,-0.1,bug),locate(0,-0.3,home), locate(2.06,2,3),locate(2.05,0,4),locate(3.06,1.1,5), locate(3.06,3.1,6),locate(0.95,3.25,7),locate(1,1,green(8)) )}}} The cube has 6 faces that can be combined in opposite pairs (front, back), (left, right), and (upper, lower) in parallel planes. Each path under the imposed conditions comprises an open loop of three sequential edges (four vertices) on one face of the cube and a similar open loop of three sequential edges (four vertices) on the opposite face, traversed in mutually opposite directions. These open loops on opposite faces are connected by connecting edges — or "bridges" — which are also traversed in mutually opposite directions. Thus, for example, in the cube 12345678 in the figure, the first open loop of three edges (four vertices) is the path 1234, while the second open loop of three edges (four vertices) is the path 5678. The first bridge is 45; the second bridge is 81 - it closes the whole path from 1 to 1. OK, very good. Now, there are 3 edges emanating from vertex 1: 12, 14, and 18. For each of these edges, there are two open loops starting from these edges: 1278 and 1234 for edge 12; 1432 and 1458 for edge 14; 1872 and 1854 for edge 18. Thus, we have 2*3 = 6 open loops listed above, created by edges 12, 14, and 18, when they are starting edges in these open loops. It creates/generates 6 possible paths satisfying the imposed conditions. The other 6 paths arise when edges 12, 14, and 18 are the bridges for the other open loops. In all, it gives 6 + 6 = 12 possible paths satisfying the imposed conditions. <U>ANSWER</U>. There are 12 possible paths for the ant satisfying the imposed conditions. </pre> My other additional lessons on miscellaneous Geometry problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-square-divided-into-three-equal-areas-by-two-parallel-lines.lesson>A square divided into three equal areas by two parallel lines</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problem-on-equiangular-but-not-equilateral-octagon.lesson>A problem on equiangular but not equilateral octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-partially-filled-cylindrical-tank-changes-position-from-horizontal-to-vertical.lesson>A partially filled cylindrical tank changes position from horizontal to vertical</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-cylindrical-glass-filled-with-water-and-then-tilted-to-a-horizon.lesson>A cylindrical glass filled with water and then tilted to a horizon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-nice-Geometry-problem-solved-using-Algebra.lesson>A nice Geometry problem solved using Algebra</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/OVERVIEW-of-my-lessons-on-additional-misc-Geometry-problems.lesson>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</A> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
