Boy or Girl Paradox

This is another interesting probability question (under the usual assumption). Here it goes.

A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male?

Fifty-fifty? Think again!

-Mit freundlichem Gruss und, Heil Schrödingers Katze!

Monty Hall Problem

There is no other question that had caused a huge uproar in USA. The question was posted to Parade magazine's "Ask Marilyn". Who is that Marilyn and what was the question you may ask. Well, Marilyn is Marilyn vos Savant, famous for being listed for years in Guiness World Records Hall of Fame as the person with the world's highest recorded IQ (228). And here is the question :

Suppose the contestants of a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After the contestant pick a door, the host, who knows what's behind all the doors, opens one of the unchosen doors, which always reveals a goat. He then says to the contestant, "Do you want to switch to the other unopened door?" Is it to the contestant's advantage to make the switch?

At face value, this appears to be a silly question. After all, there are only two choices left, it's either open the door with a car behind or a goat instead. It's fifty-fifty isn't it? However, Marilyn said that it's better to switch.

Her response caused an uproar, it brought an avalance of mail, 10,000 by her estimate. Well, what's the big deal about it? Almost 1,000 PhD holders, many are Maths Professor joined the uproar. Agreeing that there should be 50/50 chance no matter whether the participant decide to switch the door.

However, computer simulation was in favour of Marilyn, hundred of trials that come out 2 to 1 in favor of switching. Those PhD holders were fooled. So, why switching the door give the contestant an advantage?

-Mit freundlichem Gruss und, Heil Schrödingers Katze!

The story of enigmatic e

Okay, what is e in the first place? It the one of the few mathematical constant you might ever encounter. e is an irrational number, and it can't be rerepresented as a fraction. The value of e is 2.7182 8182 8459 0452 3536 0287 4713 5266 2497 7572…. Where … is a non repeating sequence. Where does this enigmatic number came from?

It turns out that it begins with money. Yes, I'm not kidding. In 17th century, Jacob Bernoulli was studying an interesting problem about compound interest. What is the problem you may ask. Well, here it is.
Let's say that a bank give you 100% interest per year (no bank will give you that much ) and you put $1.00 in your account. At the end of the year the money in your acount will be $2.00. But what if there is another bank that give 50% interest per half a year? At the end of the year your saving will be $2.25 if you put your money there. What about 33.3…% per 4 months? Your saving will be $2.37 at the end of the year. You obviously want the one that give the highest interest right? If there is another bank that offer higher frequency of interest computation, we will jump over it don't we (assuming the number of computation of interest times percentage of interest is the same)? This leads us to infinite division of interest.

So, what is the formula? It is basically f(n) = (1+1/n)^n. Where n is the number of division (assuming that the the number of computation of interest times percentage of interest is 100%. If it's not, then you need to do some manipulation).

What e has to do with this? Before I answer the question, try playing with the formula a little bit. You might notice something interesting. As you increase the value of n, the difference between f(n) and f(n-1) become smaller and smaller. It tells you that there is a limit to the value of f(n) as n approaches infinity. And what it is? It's the famous e!

-Mit freundlichem Gruss und, Heil Schrödingers Katze!

Riemannian Geometry

Okay, let's start from what is Riemannian Geometry. It is simply Geometry on curved plane or what commonly called eliptical geometry.

Just look at the picture. In the positive curvature, the sum of the angles is > 180 degree or where is the Goddamn pi π and it is smaller when it is in saddle shaped curvature.

Need a better picture? I'll get them later.

Do you know that.....

Do you know that in Riemann Geometry, sum of the angles of a triangle is not always 180°?

Explanation will come later.

Project Euler

Quite recently, I stumbled upon this so called Project Euler, named after famed German Mathematician. It is a collection of challenging mathematical questions.
Here is the description from the website:

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context.

Good luck in trying some of the questions XD

Surprising Number Pattern

1 x 8 + 1 =9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

edit : taken from the book Math Wonders to Inspire Teachers and Students.

Interesting Pattern

Have you ever noticed this?

11^2 = 121

111^2 = 12321

1111^2 = 1234321

11111^2 = 123454321

...

111111111^2 = 12345678987654321

1111111111^2 = 1234567900987654321

and so on.

edit : taken from the book Math Wonders to Inspire Teachers and Students.

How true is this?

When I was bored enough during maths lesson, I jotted down some prime number and find something rather surprising. Prime number that is larger or equal to 7 can be re-represented as the sum of three prime numbers. in short, d = a+b+c where a,b,c and d are prime number and d≥7.

Some example up to prime number that is less than 100 would be :
7=2+2+3
11=2+2+7
13=3+3+7
17=5+5+7
19=7+7+5
23=11+5+7
29=23+3+3
31=23+3+5
37=31+3+3
41=31+5+5
43=31+5+7
47=37+5+5
53=43+5+5
59=53+3+3
61=31+13+17
67=61+3+3
71=61+5+5
73=67+3+3
79=73+3+3
83=73+5+5
89=79+5+5
97=41+37+19

Enough of spamming out of the blue.

Thanks to Mr Chew for correcting the mistake.

A Quiz

There is a frog fall inside a 30 meter deep well. The frog can climb 3 meter per day and drop 2 meter per night. Assuming the frog fall on the beginning of the day, how many days needed for the frog to get out of the well.

Pascal triangle and number 11

When I was playing with number 11, I find a rather familiar number pattern for 11^x
we all know that 11^2 = 121 and 11^3 = 1331
121 and 1331 remind me of the polynomial (a+b)^x which leads me to pascal triangle.



Look at the number rows above. The first represent 11^0 which is 1, the second represent 11^1 which is 11 and soon on.
does this mean that 11^5 = 15101051? of course not. It should be calculated this way.


So, the answer is 161051

This method could be applied for 11^x for any value of x due to the nature of no.11

11 = 10+1

Just replace the x with 10 and the y with 1, we will get



since will always be one,




notice that can be derived from pascal triangle.

The last digit of 11^x will always be 1 when the second last will always be the last digit of x

edit: I just realised that the long equation is being cut. Since it does not affect that much, I'll just leave it that way.


last edited on : 15th of May 2010

Subtraction problem

Consider this subtraction problem

1241
- 587

Most people won't like to do this in their head (or even paper)

This subtraction problem could be simplified by subtracting 600 instead of 587

1241-600 = 641

But, we have subtracted too much, 13 (600-587) too much.
So we add the "missing" 13 to the 641 we have. And we get the answer which is 654.

End of Chapter 0

* If there is a positive feedback, I'll post more of this.

Tips on fast mental calculation

Hey guys, I just bought a book on mental calculation. It is called Think Like A Maths Genius, written by Arthur Benjamin and Michael Shermer. Some of the tricks are damn obvious, but I'll type it here anyway.

Chapter 0
Quick Tricks:
Easy (and impressive) Calculation

Instant Multiplication of 11

71 x 11
To solve this,, add the digits 7 + 1 = 8 in the middle of 7 and 1
71 x 11 = 781

How about
76 x 11
Similar way, 7 + 6 = 13
BUT the answer is not 7136
It should be done this way


1
736+
836

How about 3 digit numbers?
Simple. It can be done in similar way.
Eg. 123 x 11
1 + 2 = 3
2 + 3 = 5
123 x 11 = 1353

Multipying 2 2-digit number with the same first digit and second digit that sums to 10.

71 x 79
The first 2 digit will be 7 x 8 = 56
and The last 2 digit will be 1 x 9 = 9
so the answer is 5609.

*more will be added soon.