Cool Math Stuff: May 2012

Saturday, May 26, 2012

It's Math in a Card Trick!!!

I haven’t wrote about many mathematical card tricks in the last little while. The last one I remember was the guessing one where you must try to guess each card’s value incorrectly through the deck. Because of probability and the number e, you will find it difficult to get through 52 cards incorrectly.

The reasoning behind that trick is pretty hard to beat. However, that trick definitely isn’t the best trick out there. The one I want to talk about today is a tad more impressive; something that with a little practice, you will be showing to all of your friends.

Here is how the trick goes: take a seemingly mixed deck of cards and cut off a stack. Then, supposedly by feeling the weight of the cards, determine the number of cards in the cut. You can repeat the effect as many times as you like with your own variations (riffling the cards and saying you can hear each click, etcetera), and it should fool your spectators.

The method is extremely simple. It has a little mathematical reasoning behind it, and involves a small mental calculation.

First, you must arrange the deck in a specific order before the trick. You can fan the deck out and people won’t tell a pattern, but there is one. The stack goes like this:

10D

10C

10H

10S

If you look closely at this pattern, it is going up by three for each card. For the suits, you could technically do it however you want for this trick. However, it is traditionally ordered at clubs, hearts, spades, diamonds in a repeating sequence. You can remember this with the acronym CHaSeD.

Once you’ve got that done, you can begin the trick. If you know some false shuffles (the performer seemingly shuffles the deck without changing the order), you can use them. I would not recommend trying a false cut however, since cutting the deck is okay for this trick.

When you are cutting the deck, occasionally glance at the bottom card (do this while making a hand gesture or some natural movement that will not make it apparent that you made that glance). Wait until it is a higher card, and once it is, memorize that card value. Say it is the Queen of Spades.

Now, you can cut the deck yourself, or have the spectator cut it. Once you have the stack, glance at the bottom card. Let’s say that is the Nine of Spades. Here is what you have to do.

First, make sure you understand the following:

A = 1

J = 11

Q = 12

K = 13

So, the Queen of Spades is equivalent to 12. Subtract the 9 (Nine of Spades) from the 12 (Queen of Spades).

12 - 9 = 3

Now, multiply this number by 4.

3 x 4 = 12

Now, ask yourself this question: is there any way I am holding 12 cards?

You will be able to tell if you are or aren’t holding 12 cards. In this case, you would be holding 12 cards.

Let’s say the bottom card is the Jack of Hearts and you cut to the Four of Spades. Same thing; do the following calculation:

(11 - 4) x 4

7 x 4

Now, ask yourself if you are holding 28 cards. If you actually try this, you will find it clear that you are holding more than 28 cards.

Here’s what you do. With our sequence, there is a four every thirteen cards. Because of this, we can add or subtract 13 to the number we have and it won’t make a difference. So, let’s try it.

28 + 13 = 41

In this scenario, 41 would be a reasonable estimate, and the correct answer.

Let’s try one more; say the bottom card is the Jack of Diamonds and you cut to the Queen of Hearts. So, you would do:

(11 - 12) x 4

-1 x 4

-4!!!!

You can’t have -4 cards. However, we can add 13 as we please.

-4 + 13 = 9

9 + 13 = 22

This stack would have 22 cards.

It will take practice to make the glance at the bottom of the cut and the bottom of the deck more slick, and the mental arithmetic quicker, but it is definitely a cool effect. If you know false shuffles, it will become even better. But most importantly, it is a fun way to bring math into your everyday life.

Saturday, May 19, 2012

School Math Stuff on Cool Math Stuff

Most of the material I use for posts is information I find out about outside of school. I believe that these things should be taught in school. Lots of the posts I make are on algebra, primes, Fibonacci numbers, and other stuff that fit right into the lesson plans of teachers. It would only take two minutes or so to teach some of these things, and can make a huge impact on how much students enjoy math class.

Today, we run into a rare occasion where the post is on something I learned inside of school. It is a concept that I found cool, but my classmates didn’t. I think this was because it was presented to us as the method we have to use rather than a pattern that can make our life much easier.

In algebra, we run into situations where we need to plug a number into an equation, or find out what an equation equals if x equals a certain number. For instance, say you had the equation:

f(x) = x - 3

If we need to find out what this equation equals if x = 5, we do:

f(5) = 5 - 3

f(5) = 2

This was fairly easy. We just had a little subtraction. However, what if we had to plug 1 into this equation:

f(x) = x^5 - 3x^4 + 10x^3 + 2x^2 - 18x + 14

This is pretty complicated. You would have to go through this long process:

f(1) = (1)^5 - 3(1)^4 + 10(1)^3 + 2(1)^2 - 18(1) + 14

f(1) = 1 - 3 + 10 + 2 - 18 + 14

f(1) = 6

That wasn’t too bad, but imagine if that one were a two. Then we would have a problem. But there is a much easier way to do this.

1 | 1 -3 10 2 -18 14

Above, we have the number we are plugging in in that top left section. Then, we have the coefficients (the numbers before the x in the equation) lined up in order.

The first step to this method is to bring the one down below the line. Then, you multiply that one by the number in the top left. Put that directly below the -3.

1 | 1 -3 10 2 -18 14

1 _

Now, we must add the numbers in that second column to get the number that goes below. -3 + 1 = -2, so we put that below.

1 | 1 -3 10 2 -18 14

1 _

1 -2

And then, we do the same process as before. -2 x 1 = -2, and we put it below the 10.

1 | 1 -3 10 2 -18 14

1 -2 _

1 -2

Now, we add, and do the same thing.

1 | 1 -3 10 2 -18 14

1 -2 8 10 -8

1 -2 8 10 -8 | 6

The last number you get after adding is the answer. In this case, we got six (I sectioned it off), and as we saw, six is what we get after plugging one into the equation.

I find this interesting not because of any patterns or proofs, but because of its simplicity. There is another reason which is a little hard to explain, but it is related to division of polynomials. I hope it didn’t seem complicated, because it really isn’t. It is very cool after you think about it.

Saturday, May 12, 2012

The Mathematical Game of KenKen: Sudoku on Steroids

To start off, I would like to give an update on Tau 2000. This past Sunday, I established the world record my reciting from memory 2012 digits of tau, raising over $3000 dollars for the Bethel Public Library. You can find out how the event went at www.Tau2000.com, or check out this video:

A part of the Tau 2000 event was the KenKen contest, which has contestants and even finalists who had never played KenKen before this event. This was true proof to me about how simple it is to learn that I had to write a post on it.

In short, KenKen is basically Sudoku on steroids. It has the same rules as Sudoku in that you cannot repeat a number in any row or column. However, it does not start you with any numbers like Sudoku does. Rather, it gives mathematical operations to help you solve the puzzle.

Here is a video to teach you how to play KenKen done by the crossword puzzle editor of the New York Times: Will Shortz.

It's pretty simple right? And once you start trying it, it becomes addictive. Here is a sample puzzle to get you started:

I will post the answer to this puzzle in a month, as usual with puzzles. For more puzzles, you can check out www.KenKen.com.

Saturday, May 5, 2012

Fibonacci Day: Two Fun Facts

Today is Fibonacci day! It is the fifth of May, and five is a Fibonacci number.

As you may know already, I am hosting the Tau 2000 fundraiser for the Bethel Public Library tomorrow, and I am extremely busy practicing my memorization of 2000 digits of the number tau.

I usually do something that is involves some thinking to appreciate the coolness of the proof, trick, pattern, or other mathematical fun fact. However, since I am very busy, I am going to write about a few things that are just quick fun facts about Fibonacci numbers.

Several months ago, we discussed what happens when you square Fibonacci numbers. Now, I'd like to do something similar: see if there are any that are already square. Let's go through.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6725, ...

On this list, there are only three - all towards the beginning of the list. They are 1, 1, and 144. Is there anything interesting about that?

In fact, there is. These three numbers are the only square Fibonacci numbers that exist. Since I don't know how to prove this, I can't, but I'm sure it involves some very complicated math that I wouldn't understand, since I wouldn't know where to begin.

Here is another fun fact. For school subjects, math is commonly applied in science and social studies (dates, coordinates, etc.), but very seldom in language arts. This is a very cool way to relate poetry with Fibonacci numbers, two seemingly different topics.

Take a limerick, which goes something like this:

di dum di di dum di di dum
di dum di di dum di di dum
di dum di di dum
di dum di di dum
di dum di di dum di di dum

How many syllables are in each line? Coincidentally enough, each word is one syllable, which makes it easy to count.

di dum di di dum di di dum = 8

di dum di di dum = 5

di dum di di dum di di dum = 8

Is there any pattern in those numbers? In fact, there is; they are all Fibonacci numbers! Not only that, but there are 3 eights and 2 fives, and both 3 and 2 are Fibonacci numbers. It gets better! If you add up the total number of syllables, you get 34, a Fibonacci number.

We can take it even further! There are 5 di's and 3 dum's in each of the eight syllable lines - both Fibonacci numbers. In the five syllable lines, there are 3 di's and 2 dum's - two Fibonacci numbers again. There is a total of 21 di's and 13 dum's - and those are Fibonacci numbers. Though you wouldn't think it, limericks are filled with Fibonacci numbers. I thought that this was pretty cool!

Answer: A month ago, I gave you the giraffe puzzle:

To make the other giraffe, you must move its back leg over and make it parallel with the body. Then, you just rotate the picture clockwise, and you have it. At the end, it will look like this:

If you solved it, congratulations. It is another puzzle that is so simple that it's hard, which are usually the fun kind.