Strategic Practice and Homework 8 - Projects at Harvard
Strategic Practice and Homework 8 - Projects at Harvard
Strategic Practice and Homework 8 - Projects at Harvard
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3. The circumference of a circle is colored with red <strong>and</strong> blue ink such th<strong>at</strong> 2/3 ofthe circumference is red <strong>and</strong> 1/3 is blue. Prove th<strong>at</strong> no m<strong>at</strong>ter how complic<strong>at</strong>edthe coloring scheme is, there is a way to inscribe a square in the circle suchth<strong>at</strong> <strong>at</strong> least three of the four corners of the square touch red ink.4. Ten points in the plane are design<strong>at</strong>ed. You have ten circular coins (of the sameradius). Show th<strong>at</strong> you can position the coins in the plane (without stackingthem) so th<strong>at</strong> all ten points are covered.Hint: consider a honeycomb tiling as in http://m<strong>at</strong>hworld.wolfram.com/Honeycomb.html.You can use the fact from geometry th<strong>at</strong> if a circle is inscribed in a hexagon⇡then the r<strong>at</strong>io of the area of the circle to the area of the hexagon is2 p > 0.9. 33
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