Which whole numbers are expressible as sums of two (integer) squares? Here’s a theorem that completely answers the question, due...

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## Impossible Integral?

The following integral may be problematic for a freshman calculus student, even if armed with a list of antiderivatives: INTEGRAL0...

Continue reading...## Chords of an Ellipse

Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced...

Continue reading...## Sliding Chords

Take a circle C, and a chord in the circle. Now slide the chord around the circle. As you do this, the...

Continue reading...## Unbelievable Unlinking

Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber....

Continue reading...## Pythagorean Triples

Which triples of whole numbers {a, b, c} satisfy a2 + b2 = c2 ? Such triples are called Pythagorean triples because they are integer...

Continue reading...## Tower of Powers

Consider an infinite “tower of powers” of x, defined by x^x^x^… = x^(x^(x^…)) Can we find a value of x...

Continue reading...## High-Dimensional Spheres in Cubes

How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle...

Continue reading...## Rolling Polygons

Perhaps you’ve learned from a calculus class that as you roll a circular disk along a straight line, that the area under...

Continue reading...## Rubber Bands Stuck on a Torus

Consider a rubber tire (a torus) with a hole in it. Suppose that there is a green rubber band stuck...

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