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Math Puzzles, Activities, and Games in the classroom Hermany-Project, 2016
Puzzle 1: Neighboring square numbers
Problem: Write down the numbers 1-16 in a row so that the sum of two arbitraryneighbors is a square number.
Solution: The number 16 must be at an end. Only 9 can be a neighbor. Noweverything is forced. We get a solution to a Hamiltonian path problem in a squaregraph 16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8
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Subject: Arithmetic, square numbers, graph theory
Problem: Write down the numbers 1-32 in a circle so that the sum of two arbitraryneighbors is a square number.
Solution: This asks for a Hamiltonian cycle in a graph. 32 appears is the smallestnumber for which one can solve the problem. To investigate this, we can for every nwrite down a graph with vertex set {1, . . . , n} such that (a, b) are an edge if a+ b isa square. Lets call them the square number graphs Gn. Now G32 is the smallestof these graphs which is a Hamiltonian graph.
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Source: Original question 1: Anna Beliakova, Uni Zurich, und Dmitrij Nikolenkov, Highschool Trogen, They also ask for the cyclic problem with 16 numbers, where no solution exists.
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