Yes I'm Still Alive!

HEY!

Sorry I couldn't get back sooner... I tried to put up a post a couple times but it wouldn't save or publish the post. Anyhow, I'm back! I'll try to do a summary of last ones this time for yall.

Pigeonhole Principle-- the idea of putting pigeons in holes. If there are m pigeons in m holes, then––
a) either there is one pigeon per hole;
b) or there is an empty hole and a hole with more than one pigeon.

If there is m pigeons in n holes, and m>n, then there is at least one hole with more than one pigeon. If m<n, then there is at least one empty hole.

This is essentially the same thing as the socks from last week–– but the socks have magically transformed into birds (sheep or cotton plants would be much more fitting). You want at least two sheep in one of the holes-- so you make the number of sheep more than the number of holes.

Here's an interesting problem involving the pigeonhole principle that I found––


There are five points on the surface of a sphere. Prove that at least one hemisphere has four of the points in it.
In spherical geometry, any two points on the sphere determine a great circle that divides two hemispheres. According to the pigeonhole principle, at least two of the other points should be on one of the hemispheres. So, including the two on the great circle, there are four points on that hemisphere. Cool, right?


I also found a really cool logic/geometry problem in a book I read.

In a square swimming pool, there is a boy in the center. At one of the corners is a teacher that can run three times faster than the boy can swim. The boy can run faster than the teacher can. Logically, can the boy escape if the teacher must try to catch him? (Don't make assumptions!)
Because the boy is faster than the teacher on land, his goal is simply to reach the edge. If the boy tries to swim to the opposite corner, the teacher will arrive before he does. However, if he tries to swim to one of opposite edge midpoints, the teacher will arrive exactly as he does. If the teacher was a fraction slower, he would arrive before and escape. So, if he begins to travel towards one of the opposite midpoints, the teacher is forced to head there.
Consider the diagram.

If the boy swims n/3 units towards midpoint of BC, then the teacher is required to move to midpoint of AB because if they don't, the boy will escape that way. Then, the boy can swim n units towards  CD and escape. Easy as pi.

Well, I guess that was a good long one for everybody. Again, sorry I haven't posted.

STay coolio,
John

Color Coding (both types)

When you hear the words 'color' and 'coding' together, you think of moms and control freaks, right? Well, in computer coding, there's actually a way of making every single color, sprinkled right in with set theory and hexadecimals.

When you want to express a color in coding, you use the amounts of Red, Blue, and Green. Therefore they call it the RGB Color System.

In coding, there are 255 different levels of each specified color. Since 256 is the square of 16, hexadecimal is the way to go in actually expressing the colors for the computer to register, convert, and show.

For the final input, you get a six digit number with some digits represented by letters from A through H. Basically, each color gets two digits to express its level. Examples:

#FF7000 is UVA ORANGE
#FF66FF is BREAST CANCER AWARENESS PINK
#00FF00 is MY FAVORITE COLOR (try to out what color this is! Don't cheat!)



Continuing a color streak, here's a problem about colors.

Let's say there are 8 red socks (#FF0000), 10 gray socks (#888888), 12 purple socks (#9933CC), and 14 white socks (#FFFFFF) in a drawer, all individually separated into a random pile. You just need a pair of socks–– any color, as long as they match color. However, either the lights or off or you're blind because you can't see what color each sock is. How many socks do you have to grab in order to make sure you get at least one match? How about 2?

Because you have to make absolutely sure, you think of the worst possible scenario, like all the color-coding control freaks at home. The worst possible scenario is that you'd get one of each sock. However, you need one pair, so you can add one and get 5 socks. As for two, you do the same thing: one from each of three colors and three of the last. Add one to get 7.


You  want a pair of white socks, two pairs of gray socks, and a pair of red socks from that same drawer (and by the way, you put back the ones you took out). How many pairs do you have to pull out, at least?

This is basically the same idea: worst possible scenario and add one. Logically, you can get all of the purple socks without getting a correct pair at all. Since you want to have the most possible socks, you should get all of the white and gray socks too, and one red sock. That would give you 38 socks (by the way, you could also get all red, purple, and white socks and then 4 gray socks for the same answer).


Have a 'colorful' day in paradise...

Stay cool,
John

100 Pageviews!

Since my blog has hit 100 page views, I am going to do a binary special!

If you convert "John" into binary code and then back into base ten, you get '74 111 104 110', or '1,248,815,214' without letter spacing.

Some people actually get in trouble for copying distribution codes (in binary) and converting them into regular letters in order to redistribute them–– so technically, they're illegal!!!

2^x binary codes can be made x bits (binary digits) long, because there are two digit options per bit.

There are 3 main bases in the number system: binary (base 2), decimal (base 10), and hexadecimal (base 16). Hexadecimal is used to express binary numbers up to 4 bits long.

HEXADECIMAL DRUM SET–– http://www.mathsisfun.com/games/hex-drums.html
Math is Fun is a cool website to find stuff like this on

Sorry about subscribing issues, I'll try to fix that. Thanks to my sole follower, Ben K., for finding a way around that. See ya.

Stay coolio,
John

Venn Diagrams and Set Theory

I was researching Set Theory last night, and after thinking a little bit, I realized that geometric numbers could be used to represent items in a set. I was trying to figure out if you could make a Venn-type diagram to represent all of the subsets as equal areas. I figured out 4 sets...

but I couldn't find out what the pattern was until later. Let's say you are making a Venn-type diagram out of X sets. There would have to be all possible intersections included, or 2^X different areas. So, if 2^X is any kind of geometric number, then it can be constructed from that (regular?) geometric shape. Just a little theory...

Set theory is like the coolest thing ever. It's like all of logic in one tiny part of math. It's important for probability, exponents, and number theory, of other things. It's the idea that we can group everything, rearrange, them, and still  have the same thing. How cool is that?

The most basic functions of sets are complements (what isn't in a set), unions (the combination of sets), intersections (what two sets have in common), and if (or if and only if) statements. Combining functions and sets, you can separate out any part.


That's pretty cool.

STay coolio,
John

Problems of the Week

Here's some cool problems that I saw.

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In a square multiplication table with the numbers 1 through 9, what is the sum of all of the products within the table?

If you think about it, the sum of each column x is 1x+2x+3x+4x+5x+6x+7x+8x+9x. All of the columns are numbered 1 through 9, so by distributive property, the sum of all the numbers is
(1+2+3+4+5+6+7+8+9)*(1+2+3+4+5+6+7+8+9), or 2025. Just a cool thought.


In some languages, every consonant must be followed by a vowel. How many words can be made from the letters in the Hawaiian word 'MAKAALA' if every consonant must be followed by a vowel?

If you think about it, the letters can be separated into four separate units 'MA', 'KA', 'LA', and 'A' because each consonant must be followed by an A. You can arrange the units in 4!=24 different ways.


Let set S be the set of all positive integers that have a remainder of 12 when divided into 192. What is the median of the set?

There are two rules to the set––
1. S must be greater than 12, by definition.
2. S be a factor of 180 (because 192-12=180).
If you write out the union of the two sets above, you find the median is 36. Just simple set theory.


There are two sticks of lengths A and B, where B is greater than A. To make a triangle from these two sticks and one extra stick, the extra stick must be strictly between 8 cm and 26 cm long. What is the area of a rectangle with side lengths A and B?

This is a cool one for its geometric properties. By definition, any one of a triangle's sides can be no longer than the sum of the other sides. So, we can work it down to––
A+B<26 and B<A+8
and from solving, A and B are 9 and 17, respectively.

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We should make that a thing, where I tell you a bunch of cool problems and stuff! Fun math is the best kind of math.

Stay coolio,
John

PS.–– NEXT UP: SET THEORY!!!!!

Golden Ratio

The Golden Ratio is equal to  (1 + sqrt(5))/2. Here are some interesting properties of the Golden Ratio.

If you took a rectangle bounded by the Golden Ratio (meaning, with sides 1.61803...x and x where x is not zero) and cut the biggest square you could make off of it, you'd get a rectangle similar to the original. Basically,
1.618... : 1   =   1 : (1.618... - 1)

Artwork that incorporates the Golden Ratio in its dimensions is known to be 'visually pleasing'.

The inverse of the Golden Ratio is equal to the difference between it and one. The inverse is also equal to is conjugate.

Let's say the Fibonacci sequence is represented by F(x).
F(infinity) / F(infinity-1) = 1.618...

Also, 

Which I thought was pretty darn cool.

The Golden Ratio is used by many plants to get more sunlight, as the angle it forms spreads out the most easily.

The Golden Ratio is a root of some simple quadratic functions, like x^2-x-1.

Just thought that was cool.

Stay coolio,

John