A repunit is a whole number consisting of only 1’s, such as 1, 111, or 111111111. (It’s like a repeating unit.) These numbers have some fun properties. For instance, many have already noted that the square of a repunit exhibits a nice pattern: 111*111 = 12321,1111*1111 = 1234321,11111*11111 =...

Continue reading...# easy

# Sine of (1/55555….)

Take your calculator, and enter a number consisting of several 5’s, such as 5555555. Now take the reciprocal of this number. Now take the sine of this result (in degrees). You should get a very interesting number: 3.14159… x 10-9. Are you surprised? Presentation Suggestions:The result is quite striking. (If...

Continue reading...# Divisibility by Eleven

It is easy to tell that the following are multiples of 11: 22, 33, 44, 55, etc. But how about: 2728, or 31415? Are they divisible by 11? Here an easy way to test for divisibility by 11. Take the alternating sum of the digits...

Continue reading...# Behold! the Pythagorean Theorem

Figure 1 shows one of the simplest proofs of the Pythagorean Theorem. It is also perhaps the earliest recorded proof, known to ancient Chinese, as evidenced by its appearance in the classical Chinese text Zhoubi Suanjing (compiled in the first centuries BC and AD). However, the Pythagorean theorem...

Continue reading...# Why Does 0.999… = 1?

Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of nines: 0.99999… It may come as a surprise when you first learn the fact that this real number is actually EQUAL to the integer 1....

Continue reading...# Area of a Hyperbolic Wedge

The unit circle can be parametrized by (cos w, sin w). Given a point on it, the region cut out by this circle, the x-axis, and the ray from the origin to this point has area w/2. As you may know, the hyperbolic cosine and...

Continue reading...# tamreF’s Last Theorem

Here’s an amusing theorem that is very easy to prove, which we’ll call tamreF’s Last Theorem. In fact it is tamreF’s only known theorem. It says: The following equation Nx + Ny = Nz has no solutions in positive integers for N greater than 2. This theorem has not been...

Continue reading...# Hugging the Equator

Suppose that you have a rope around the equator of a basketball. How much longer would you have to make the rope so that it is 1 foot from the surface of the basketball at all points? The answer is 2*Pi feet. Now suppose you...

Continue reading...# Gaps in Primes

We know there are infinitely many primes, so are many interesting questions you can ask about the distribution of primes, i.e., how they spread themselves out. Here is something to ponder: are there arbitrarily large “gaps” in the sequence of primes? At first this may seem like...

Continue reading...# Football Field

Take a long rope, tie it to the bottom of the goal post at one end of a football field. Then run it across the length of the field (120 yards) to a goal post at the other end. Stretch it tight, and then tie...

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