MathIsEverything
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- Sep 17, 2022
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A triangular tile is called a triangle with sides of lengths 1, 1, sqrt3, in which a line connecting the centers of the sides of length 1 is marked. A six-sided tile is called a regular hexagon with side length 1, in which three lined connecting the centers of (next to) sides are marked. Figure out all integers n ≥1 with the following property:
From 3n triangular tiles and 1/2(n - 1)n six-sided tiles, an equilateral triangle with side n*sqrt3 can be arranged in such a way that all marked lines form a single closed curve. The tiles can be rotated.
I made some progress drawing some possibilities and came to the result that n must be of the form 4k+1 or 4k+2, where k is an integer bigger or equal 0. In other cases there is one closed part/section that cannot be connected to the rest of the curve, but im not really sure how to prove my explanation (of why something works or happens the way it does). Thanks for any help
From 3n triangular tiles and 1/2(n - 1)n six-sided tiles, an equilateral triangle with side n*sqrt3 can be arranged in such a way that all marked lines form a single closed curve. The tiles can be rotated.
I made some progress drawing some possibilities and came to the result that n must be of the form 4k+1 or 4k+2, where k is an integer bigger or equal 0. In other cases there is one closed part/section that cannot be connected to the rest of the curve, but im not really sure how to prove my explanation (of why something works or happens the way it does). Thanks for any help
