easy

These fun facts are at the easy level.

Irrationality by Infinite Descent

The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the...

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Computability of Real Numbers

We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We...

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Van der Waerden Theorem

Can you color the integers red and blue such that there are no monochromatic arithmetic progressions (AP’s) extending infintely in...

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Rational Irrational Power

If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to...

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Sum of Cubes and Beyond

We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the...

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Odd Numbers in Pascal’s Triangle

Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N...

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Pretzel Unlinking

Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and...

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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...

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Pythagorean Triples

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

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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...

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