Invisible Dice Trick

Today happens to be the fifth Saturday of the month. Since this doesn't happen very often, I decided to treat you with a little mathematical magic trick.

I first found this trick on the show ScamSchool, where I also found out about Benford's Law. Rather than explaining the trick, you can go watch the video and then I will explain some of the mathematics behind it. Click here for the episode.

This trick requires a little mental math, and also a little algebra to prove to work. It is extremely basic algebra though, so don't get too worried.

With algebra, you denote unknown quantities with a letter. In this example, the unknown quantities are the values of the two dice. We can denote them with a and b. Assume that a is the one that they pulled back.

The instructions were to first multiply a by 2.

a • 2
2a

Next, the spectator had to add five.

2a + 5

Then, the spectator multiplied this quantity by five. This requires the distributive property.

(2a + 5) • 5
(2a) • 5 + (5) • 5
10a + 25

Finally, the person was asked to add the value of the other die, which we called b.

10a + 25 + b
(10a + b) + 25

After rearranging the numbers, you can see the breakdown. If you know that multiplying a number by ten gives you the same number with zero tacked on, you will realize that 10a+b is just a number with a as the first digit and b as the second digit.

So, we have the number that tells us a and b, and this is added to 25 to get the grand total. That is why the performer must subtract 25 to find this number.

I think that this trick is a super simple way to apply mathematics to magic.

Srinivasa Ramanujan

Today, I would like to tell a story about an Indian mathematician named Srinivasa Ramanujan. He was born in 1887 and died in 1920. Even though he had such a short life, he is said to have been one of the best math minds of all time.

Because of the poverty in Southern India at the time, his math education was restricted to two books: Plane Trigonometry by S.L. Loney and Synopsis of Elementary Results in Pure Mathematics by G.S. Carr. As a young boy, Ramanujan read and interpreted the information in these books, and began to rediscover numerous mathematical concepts.

He ended up sending some of his discoveries to three British mathematicians. Two of them thought his work was disorganized and his ideas were unrealistic and far-fetched. The third one, G.H. Hardy, was skeptical, but  realized that even though the work was sloppy, it was ingenious. He invited Ramanujan to come study with him in England.

Despite Ramanujan's religious beliefs and mother's wishes, he ended up traveling to England. Since he had never been out of India, he hated the European lifestyle. Because of his discomfort, he ended up getting extremely sick, resulting in his early death.

Even though this story is pretty sad, there is a more well-known, amusing story from his life. When Ramanujan was in England, Hardy went to visit him in the hospital. When he got there, Hardy had mentioned that the number on the taxi he rode in was 1729. "Rather a dull number, wouldn't you say?" he said.

Ramanujan smiled, and responded, "No, Hardy, not at all. 1729 is a fascinating number! It is the smallest number that can be expressed as the sum of two cubes in two different ways."

If you were to add together 2³ and 3³, you would get 35. However, you cannot find two other cubes to add together to get this.

However, Ramanujan noticed the number 1729 could be derived with two different pairs of cubes.

10³ + 9³ = 1000 + 729 = 1729
12³ + 1³ = 1728 + 1 = 1729

I thought this was an interesting little fact, and a fascinating backstory to go with it.

Benford's Law

Take any random quantity. Maybe a population of a city, a mass of a planet, a distance from a star, or an amount of twitter followers. Now, take the first digit of this number. What are the odds that it will start with a 1, or a 2, and so on?

One would think that it would be a one in nine chance for each digit. If you are randomly selecting this number, the first digit is just a random number selection between one and nine.

However, this reasoning does not work. Try finding these quantities yourself. Or, just go to http://testingbenfordslaw.com. If you check all of the different areas that are possible, you will see that the smaller the digit, the more commonly it appears.

Here are the approximate odds for each digit:

1: 30.1%
2: 17.6%
3: 12.5%
4: 9.7%
5: 7.9%
6: 6.7%
7: 5.8%
8: 5.1%
9: 4.6%

Since this is a math blog, something we would want to do is find a pattern between these numbers. This could probably be done on a graphing calculator, using similar techniques to the post on data analysis.

It is clear that a line of best fit would not be the solution. If we connected the points with a curved graph, it would look like this:

You can see that this graph gets really close to the y-axis, but it does not seem to touch it. Similarly, it gets really close to the x-axis, but doesn't touch it. Thus, the x and y-axes would be called asymptotes of this graph.

But, an asymptote could be found in a rational function, radical function, exponential function, logarithmic function, hyperbolic function, trigonometric function... So, it is hard to define this graph solely based on the presence of asymptotes.

The function that does work with this graph is a logarithmic one. A logarithm is basically the opposite of an exponent. For instance:

5² = 25
log₅(25) = 2

Just knowing that the equation is logarithmic doesn't seem to narrow it down a lot, because there are so many different types of logarithms. However, there are three types which are seen the most frequently. In fact, most scientific calculators contain just these three types.

Logarithm Type	Base	Simple Notation	Standardized Notation	Applications
Natural Logarithm	e (the irrational number ≈ 2.71828)	loge(x)	ln(x)	Calculus, Statistics, Physics, Chemistry, Economics
Common Logarithm	10	log10(x)	lg(x)	Algebra, Engineering, Geology, Spectroscopy, Music
Binary Logarithm	2	log2(x)	lb(x)	Discrete Mathematics, Computer Science, Information Theory

If we confine the possible equations to one of these, it narrows it down a lot. And this assumption is correct. The equation that the graph fits is:

y = lg(1 + 1/x)

(Remember that x is the starting digit and y is the percentage for that digit)

I was pretty surprised about this equation. But, all of the percentages listed above are the outcome of this equation.

Benford's Law can be used in biology, accounting, law, economics, etc. However, a more fun way to use it is to turn it into a game.

Tell someone that you will get the numbers 1, 2, and 3 and they will get the numbers 5, 6, 7, 8, and 9 (and nobody gets 4). You then have them come up with random quantities that they wouldn't know, and you look up the number (you can use Google or WolframAlpha for this). Every time it is a 1, 2, or 3, you win a point and every time it is a 5, 6, 7, 8, 9, they get a point. They think they have around a 2:1 advantage over you, but you are really the one with the 2:1 advantage.

Click here to see this game played on the show Scam School.

What Sounds the Coolest? Pi or Tau

You may have noticed that today is Pi Day (or Half-Tau Day). Because of the special occasion, I thought I should post something. Since π ≈ 3.14159, I decided to set the post to go up at 1:59.

I have posted numerous times about the Pi vs Tau debate. All of the reasonings involved mathematics. However, it is fun to analyze completely non-mathematical representations of these numbers.

There have been videos posted on YouTube of a musical representation of both pi and tau. If you want to celebrate pi day without a mathematical burden, listen to these and decide which number sounds cooler.

What Pi Sounds Like

What Tau Sounds Like

Comment which number you think sounds the best!

The Museum of Mathematics

I have been trying to find a news story once a month that pertained to mathematics that I could discuss. Last month, I ended up using a website I found, but this time, I actually got a news story. On Sunday, I was told by multiple people that the Museum of Mathematics was on CBS, which was exciting for me. Click here to see the news story.

I have met Glenn Whitney on a few occasions, including at Gathering 4 Gardner in Atlanta (click here to see a post I did about that experience) and the World Science Festival Street Fair in New York City. Each time I see him there, they have set up their Math Midway, which features many fascinating ways to have fun with mathematics, including the tricycle with square wheels and a machine that solves quadratic equations by the crank of a lever.

I visited MoMath on December 27th, which might have been a mistake on my part because it was packed with people. However, I was happy at the same time that a whole room of people were going to walk out seeing mathematics in a new light.

On my blog, my posts are mainly about pure mathematics. Pure mathematics analyzes math simply for the sake of analyzing it, while applied mathematics looks for its applications to science, engineering, finance, and society. I do have many posts labeled Practical, but even these are strictly mathematical. I never introduce science and engineering concepts (mainly because I don't know them).

MoMath gave me the opportunity to learn how math can be applied to other fields. I have ridden the square wheeled tricycle, which is a way to apply mathematics to transportation, in a fun and creative way. Using some geometry, you can find a path for any shaped wheels, which I found pretty cool.

If you are interested in pure mathematics, that is what this blog and many others cater to. If you are interested in applied mathematics, go to MoMath. It really is a fun way to look at math that you never would have done before.

Why Does πr^2 Work?

Recently, I have been alluding to the pi vs tau argument, which is debating the proposal of replacing pi with tau (the equivalent of 2π). Since both sides have pretty convincing points, it is a fun thing to talk about.

Pi fans argue that by changing pi to tau, we would ruin the formula for the area of the circle. It would go from:

A = πr^2  –>   A = 1/2τr^2

As you can see, the formula looks a lot sloppier.

Tau-ists rebut this by saying that the proof of this formula requires one to multiply 1/2 by 2πr^2, thus proving the significance of 2π. I mention this a lot, but I have never actually proven it.

Rather than writing out the proof, I think it would make more sense to watch this YouTube video. It made it a lot more interesting for me.

I find this proof fascinating on its own. Also, you may have noticed when the 2π was present and was cancelled out by the 1/2. For the pi vs tau argument, this specific instance can fall to either side.