# Cool Facts

## Using Fibonacci for mile <--> km conversion

The Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
can be used to convert miles to kilometers (and vice versa). For example, to find how many km 5 miles is, take the next Fibonacci number which happens to be 8. Thus, 5 miles is approximately 8 km. Similarly, 8 miles is approximately 13 km, and so on.

This works because the growth rate of the Fibonacci numbers converges to the golden ratio (approx 1.618) which happens to be very close the km/mile conversion (1 mile = 1.609 km).

## Prime number trick (with solution)

Using prime numbers, you can amaze your friends with a prime prediction…

1. Ask your friends to pick any prime number greater than 3.
2. Square it.
4. Divide by 12.

Without knowing which prime number your friends picked, you can still tell them:
There will be a remainder of 3.
But HOW does it work?
Let’s do an example:
13 is a prime number, squaring it gives 169, adding 14 gives 183 which has a remainder of 3 upon division by 12.

This works for every prime number greater than 3, but how exactly does it work?

The mathematics behind this is rather simple.
1. Let p be a prime number, p > 3.
2. Squaring gives:
p^2.
p^2 + 14
4. Taking it modulo 12 gives:
(p^2 + 14) mod 12

We want to show that:
(p^2 + 14) mod 12 = 3
This is equivalent to:
p^2 – 1 is divisible by 12.
That is:
(p-1)(p+1) is divisible by 12.

For a number to be divisible by twelve, it has to be divisible both by 3 and by 4. We know that, out of p-1, p and p+1, one of them must be
divisible by 3; and it can’t be p, because p is prime and greater than 3. Thus, either p-1 or p+1 is divisible by 3, and so their product is also:
(p-1)(p+1) is divisible by 3.

Now, since p is a prime greater than 3, we know that it is odd. Therefore, both p-1 and p+1 are even numbers. The product of two even numbers is divisible by 4, so:
(p-1)(p+1) is divisible by 4.

Combining this with the above, we get that:
(p-1)(p+1) is divisible by 12.
And hence:
(p^2 + 14) mod 12 = 3

I. Input shoe size

II. blah

III. blah

IV. blah

V. blah (Arbitrary number that makes this work, but only for the year 2012)

VI. Input age

Output is your shoe size and age. If you input someone else’s values, it outputs their values back. Magic!

Thanks to Nick de Vera for this submission!

## 11 Magnificent Wonders of the Ice World

Not a math post, but cool nonetheless:

In polar and other cold regions there are ice, snow and water formations that are unusual, unique, and some of them so beautiful to breathtaking. Most of these wonders of nature can be visited only by scientists and rare adventurers who are ready for significant physical and financial exertions. Because of their volatility and locations, these formations can be seen only at certain periods of the year.

Thanks to Bole982 for this submission!

## A Theorem About 1 and 0

Well, I don’t know if this is known already, so if it is, please comment the name of the theorem. If it isn’t, then someone put a comment on what it should be called. A number with all of its digits less than 2 (which means only 0 and 1) and has less than 10 digits, multiplied with a number with the same characteristics has a product reverse of that of another set of two numbers with the same digits as the first set but has its digits reversed. Example below.
111111111 x 111111101 = 12345677876543211
111111111 x 101111111 = 11234567877654321

Thanks to Kirk Bamba for this submission!

## 15 Really Odd Geographical Facts

Not really math, but pretty neat!

Our planet is filled with many wonderful geographical and geological anomalies and mysteries. So many so, that we may never truly unlock all of the secrets that nature has tucked away. This is a list of 15 of the more unusual or outright bizarre facts relating to geography, geology, and the earth.

Thanks to Bole982 for this submission!

## Game Theory Paradox Explained: Losing Strategy that Wins

Do you think playing two losing games can result in a win streak? Parrando’s paradox proves this can happen in a very simple way.

The paradox is illustrated by two games played with coins weighted on one side so that they will not fall evenly by chance to heads or tails.

In game A, a player tosses a single loaded coin and bets on each throw. The probability of winning is less than half. In game B, a player tosses one of two loaded coins with a simple rule added. He plays Coin 1 if his money is a multiple of a particular whole number, like three.

If his money cannot be divided by the number three, he plays the Coin 2. In this setup, the second will be played more often than the first.

Both are loaded, one to lose badly and one to win slightly, with the upshot being that anyone playing this game will eventually lose all his money.

“Sure enough,” Dr. Abbott said, when a person plays either game 100 times, all money taken to the gambling table is lost. But when the games are alternated — playing A twice and B twice for 100 times — money is not lost.

It accumulates into big winnings. Even more surprising, he said, when game A and B are played randomly, with no order in the alternating sequence, winnings also go up and up.

## 306 years since the birth of Leonhard Euler

Today marks the 306th birthday of famous Swiss mathematician, Leonhard Euler (1707-1783), as you might have noticed from the unique Google logo. Euler made a significant number of discoveries in various fields, from infinitesimal calculus to graph theory. He is well known for introducing the notion of a mathematical function and his works take up between 60-80 quarto volumes. Here are just some of his accomplishments.

# Eulerâ€™s formula

Euler’s formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then

v âˆ’ e + f = 2.

# Seven Bridges of KÃ¶nigsberg

The Seven Bridges of KÃ¶nigsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology.

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution. There could be no non-retracing the bridges. The difficulty was the development of a technique of analysis and of subsequent tests that established this assertion with mathematical rigor.

# Euler’s identity

In analytical mathematics, Euler’s identity (also known as Euler’s equation), named for the Swiss mathematician Leonhard Euler, is the equality

where
e is Euler’s number, the base of natural logarithms,
i is the imaginary unit, which satisfies i2 = âˆ’1, and
Ï€ is pi, the ratio of the circumference of a circle to its diameter.

## Anscombe’s quartet

Anscombe’s quartet comprises four datasets that have nearly identical simple statistical properties, yet appear very different when graphed. Each dataset consists of eleven (x,y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data before analysing it and the effect of outliers on statistical properties.

The first scatter plot (top left) appears to be a simple linear relationship, corresponding to two variables correlated and following the assumption of normality. The second graph (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear, and the Pearson correlation coefficient is not relevant. In the third graph (bottom left), the distribution is linear, but with a different regression line, which is offset by the one outlier which exerts enough influence to alter the regression line and lower the correlation coefficient from 1 to 0.816. Finally, the fourth graph (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

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