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Multiplication by 111

If you liked the Fun Fact Multiplication by 11, you’ll enjoy seeing how to take that idea one step farther. Here’s a quick way to multiply by 111. To multiply a two-digit number by 111, add the two digits and if the sum is a single...

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i to the i is a Real Number

If you are familiar with complex numbers, the “imaginary” number i has the property that the square of i is -1. It is a rather curious fact that i raised to the i-th power is actually a real number! In fact, its value is approximately 0.20788....

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Face Derivatives and Computer Vision

One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand” the data it receives from some visual sensor? For example, one aspect of this problem is object recognition, and another is object tracking. While recognition is a...

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Square Root of Two is Irrational

An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to the ancient Greeks that there were lengths that could not be expressed as a fraction. For instance, they could show that a right triangle whose...

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Finding the N-th digit of Pi

Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:Pi = SUMk=0 to infinity 16-k [ 4/(8k+1) – 2/(8k+4) – 1/(8k+5) – 1/(8k+6) ].The reason this pi formula is so interesting is because it can be used to...

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Ellipsoidal Paths

Given an ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a counterclockwise fashion (say), follow a line tangent to the inner ellipse until you hit the outer ellipse again. Repeat. Figure 1 shows an example. Now...

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Large Counterexample

A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is said to be of odd type. Examples: 4=2*2 is even type, 18=2*3*3 is odd type. (We say 1 has 0 primes and is therefore of even...

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Whiffle Ball

Does a ball take longer to come down than go up, or does it take the same amount of time either way? Of course, in the absence of air friction, a ball takes the same amount of time either way. But what if air friction...

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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 made of very flexible and stretchy rubber. Question: is it possible to deform one set of links into the other in a continuous motion (without...

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