By A. Nicolaides

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What curves result if instead we look for the set of points equidistant from a dot and a circle? ) Draw two objects on a page, say, a dot and a square, a circle and an ellipse, or two circles of different radii. What curves emerge as you investigate the set of all points equidistant from your two objects? What can be said about equidistance for three objects? COMMENTARY, SOLUTIONS and THOUGHTS Here’s how to trisect an angle with origami ([HULL1]). As in the newsletter we assume that the angle is deﬁned by a line L drawn from bottom left corner A of a piece of paper and the bottom edge of that paper.

Problem 3173, American Mathematical Monthly, 33 (3), (1926), 159. 6 Factor Trees PUZZLER: Factor Trees In grade school students draw factor trees. Here is a tree for 36,000: 36000 40 900 9 3 100 3 20 10 2 8 5 5 2 2 4 2 2 5 At each stage we split the number at hand into a pair of factors, halting at the primes. (This forces the tree to stop. ) The tree allows us to write the starting number as a product of primes. Here we see 36000 = 2 · 2 · 2 · 2 · 2 · 3 · 3 · 5 · 5 · 5. What is astounding (though most people don’t seem to think so) is that despite possible different choices we can make along the way, this produces the same list of primes.

A dozen questions about pile splitting, Math Horizons, September 2004, 28–31. 7 Folding Fractions and Conics PUZZLER: Trisecting an Angle Sally draws a straight line from the bottom left corner of a blank piece of paper. She challenges Terrell to make a crease in the paper that bisects the angle formed (that is, cuts that angle exactly in half ). “Easy” says Terrell as he lifts up the bottom edge, aligns it with the straight mark, and folds. ” It can be done using nothing more than creases in the paper.