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LOGO Challenge 5 - Patch

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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LOGO Challenge 6 - Triangles and Stars

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

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Hexagon Cut Out

Weekly Problem 52 - 2012
An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

Sharks Are Symmetric

Stage: 3 Short Challenge Level: Challenge Level:1

Taking the altitude shown, symmetry can be obtained by shading 3 more triangles. Taking the vertical altitude would require 5 more to be shaded and the other sloping alitute would need 6 more.

This problem is taken from the UKMT Mathematical Challenges.