Topology can determine if equation or simultaneous equations have solutions {topological problems}. Function must be continuous, compact, and connected. If winding number is greater than zero, two simultaneous equations have solutions.
Can one cross all seven bridges of old Königsberg only once and return to start {Königsberg bridges problem}? Königsberg had seven bridges and two river islands.
Euler replaced bridges by lines and land by points to show that one path cannot traverse the bridges. Königsberg bridges problem is a topological graph, in which land is nodes and bridges are connections. If all nodes are even, one can cross all bridges once and return to start. If one or two nodes are odd, one can cross all bridges once but end at another point. If more than two nodes are odd, one must cross at least one bridge more than once.
Games {tower of Hanoi} can use a horizontal board with three vertical pegs. Discs are on one peg, with largest on bottom. The idea is to transfer discs to another peg to recreate same series, without ever putting a larger disc on a smaller-disc top. With n discs, it requires 2^n - 1 transfers.
Salesman wants to travel shortest distance among cities, with no path duplication {traveling-salesman problem}.
Four colors can color two-dimensional maps so no two countries with common boundary have same color {map problem} {four-color theorem}|. Kenneth Appel and Wolfgang Haken proved the four-color theorem [1976], by enumerating possibilities and then checking all by computer. One-dimensional line maps need two colors. Three-dimensional maps need six colors.
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Date Modified: 2022.0225