Maths, quite bored isn't it? Well maths can be more interesting that you though and there's alot of secrets things about maths you might not known. Let me make you less boring by giving.you my facts about maths.
1. The Four Lucky One
Did you know there are only four numbers(other than 1) which are the sum of the cubes of their digit
("^" means Power Of / Raise to Power)
153 = 1^3 + 5^3 + 3^3
370 = 3^3 + 7^3 + 0^3
371 = 3^3 + 7^3 + 1^3
407 = 4^3 + 0^3 + 7^3
Try finding any other possible number, this is the only four number that are possible.
2. It's All Logic
I'll tell you, maths is one of the only areas of knowledge that can objectively be described as “true,” because its theorems are derived from only logic. YES... LOGIC. And yet, at the same time, those theorems are often extremely strange and counter-intuitive but still base on logic. So maths is just, well.. think logic. How hard it can be? Right?
3. Number 37
So what's so interesting about number 37? well if you do like this:
111 ÷ 37 = 3
So 111 you divided by 37, you will get 3 which is also the sum of 111 which is 3. Try divided 222 to 999 by 37 and you'll get the same result:
222 ÷ 37 = 6
333 ÷ 37 = 9
444 ÷ 37 =12
555 ÷ 37 = 15
666 ÷ 37 = 18
777 ÷ 37 = 21
888 ÷ 37 = 24
999 ÷ 37 = 27
More surprisingly the answer is base on multiple of 3. So let me explain deeper.
111 can only be divided by 1, 3 and 37, if you try this:
1 x (3 x 37) you will get 111 having triple 1, replace 1 with 2 until 9 and you will get three of the same digit.
Well it's hard to explain how exactly this result happen but you can figure it out how.
4. Letter A
In school we learn how to spell every numbers, but do you know that the letter A only appear on the number 1000(One Thousand) and so on. A never appear from 0 to 999, good luck finding it.
The answer, of course, depends on what kind of shuffle you consider. Two popular kinds of shuffles are the random riffle shuffle and the overhand shuffle. The random riffle shuffle is modeled by cutting the deck binomially and dropping cards one-by-one from either half of the deck with probability proportional to the current sizes of the deck halves. In 1992, Bayer and Diaconis showed that after seven random riffle shuffles of a deck of 52 cards, every configuration is nearly equally likely. Shuffling more than this does not significantly increase the "randomness"; shuffle less than this and the deck is "far" from random.
That's all for today, hope you like it and thanks for reading.
Did you know there are only four numbers(other than 1) which are the sum of the cubes of their digit
("^" means Power Of / Raise to Power)
153 = 1^3 + 5^3 + 3^3
370 = 3^3 + 7^3 + 0^3
371 = 3^3 + 7^3 + 1^3
407 = 4^3 + 0^3 + 7^3
Try finding any other possible number, this is the only four number that are possible.
2. It's All Logic
I'll tell you, maths is one of the only areas of knowledge that can objectively be described as “true,” because its theorems are derived from only logic. YES... LOGIC. And yet, at the same time, those theorems are often extremely strange and counter-intuitive but still base on logic. So maths is just, well.. think logic. How hard it can be? Right?
3. Number 37
So what's so interesting about number 37? well if you do like this:
111 ÷ 37 = 3
So 111 you divided by 37, you will get 3 which is also the sum of 111 which is 3. Try divided 222 to 999 by 37 and you'll get the same result:
222 ÷ 37 = 6
333 ÷ 37 = 9
444 ÷ 37 =12
555 ÷ 37 = 15
666 ÷ 37 = 18
777 ÷ 37 = 21
888 ÷ 37 = 24
999 ÷ 37 = 27
More surprisingly the answer is base on multiple of 3. So let me explain deeper.
111 can only be divided by 1, 3 and 37, if you try this:
1 x (3 x 37) you will get 111 having triple 1, replace 1 with 2 until 9 and you will get three of the same digit.
Well it's hard to explain how exactly this result happen but you can figure it out how.
4. Letter A
In school we learn how to spell every numbers, but do you know that the letter A only appear on the number 1000(One Thousand) and so on. A never appear from 0 to 999, good luck finding it.
5. Pi
Pie is a delicious food that can made from... wait!! Not that Pie, It's Pi, the ratio of the circumference to the diameter of a circle. Anyway do you know that Pi can't be expressed as a fraction, making it an irrational number. Also it never repeats the same patten and never ends when written as a decimal. Just try counting it as many as you can and see if there is any repeated patten if you have enough paper by the way.
Yea.. just keep counting.
6. What comes after a Million?
Start from Million, Milliard, Billion, Trillion, Quadrillion, Quintillion, Sextillion, Septillion, Octillion, Nonillion, Decillion, Undecillion, Duodecillion, Tredecillion, Quattuordecillion, Quindecillion, Sexdecillion, Septendecillion, Octodecillion, Novemdecillion, Vigintillion and Centillion. Yea that's all
and also not forget Googol and Googolplex.
7. Google
So what's wrong with Google? We know that Google is the most popular search engine in the world but did you know that the word "Google" is from a misspelling of the word 'googol', which is a number one followed by one hundred zeros. Basically the name was use to show how search engine can search anything. Man no wonder Google have everything.
8. Seven Shuffles and Rising Sequences in Card Shuffling
How many shuffles does it take to randomize a deck of cards?
The answer, of course, depends on what kind of shuffle you consider. Two popular kinds of shuffles are the random riffle shuffle and the overhand shuffle. The random riffle shuffle is modeled by cutting the deck binomially and dropping cards one-by-one from either half of the deck with probability proportional to the current sizes of the deck halves. In 1992, Bayer and Diaconis showed that after seven random riffle shuffles of a deck of 52 cards, every configuration is nearly equally likely. Shuffling more than this does not significantly increase the "randomness"; shuffle less than this and the deck is "far" from random.
In fact, is that five random riffle shuffles are not enough to randomize a deck of cards, because not only is every configuration not nearly equally likely, there are in fact some configurations which are not reachable in 5 shuffles!
To see this, suppose (before shuffling) the cards in a deck are arranged in order from 1 to 52, top to bottom. After doing one shuffle, what kind of sequences are possible? A moment's reflection reveals that only configurations with 2 or fewer rising sequences are possible. A rising sequence is a maximal increasing sequential ordering of cards that appear in the deck (with other cards possibly interspersed) as you run through the cards from top to bottom. For instance, in an 8 card deck, 12345678 is the ordered deck and it has 1 rising sequence. After one shuffle,
16237845
is a possible configuration; note that it has 2 rising sequences (the Blue numerals form one, the Red numerals form the other). Clearly the rising sequences are formed when the deck is cut before they are interleaved in the shuffle.
So, after doing 2 shuffles, how many rising sequences can we expect? At most 4, since each of the 2 rising sequences from the first shuffle have a chance of being cut in the second shuffle. So the number of rising sequences can at most double during each shuffle. After doing 5 shuffles, there at most 32 rising sequences.
But the reversed deck, numbered 52 down to 1, has 52 rising sequences! Therefore the reversed deck cannot be obtained in 5 random riffle shuffles!
So the maths behind this fact is the analysis of shuffling involves both combinatorics, probability, and even some group theory. Though we've been discussing "random" riffle shuffles in this Fun Fact, you can also study the mathematics of Perfect Shuffles, which have no randomness.
Huh.. so many explanation.
Are there infinitely many primes?
I'll give a proof to show that there must be infinitely many primes. We will show that if there were only finitely many primes, it would lead to a contradiction.
So, it's common that a bike's wheels is round, right?. But do you ever imaging riding with a square wheels? Are you crazy? It's impossible to ride with square wheels.
To see this, suppose (before shuffling) the cards in a deck are arranged in order from 1 to 52, top to bottom. After doing one shuffle, what kind of sequences are possible? A moment's reflection reveals that only configurations with 2 or fewer rising sequences are possible. A rising sequence is a maximal increasing sequential ordering of cards that appear in the deck (with other cards possibly interspersed) as you run through the cards from top to bottom. For instance, in an 8 card deck, 12345678 is the ordered deck and it has 1 rising sequence. After one shuffle,
16237845
is a possible configuration; note that it has 2 rising sequences (the Blue numerals form one, the Red numerals form the other). Clearly the rising sequences are formed when the deck is cut before they are interleaved in the shuffle.
So, after doing 2 shuffles, how many rising sequences can we expect? At most 4, since each of the 2 rising sequences from the first shuffle have a chance of being cut in the second shuffle. So the number of rising sequences can at most double during each shuffle. After doing 5 shuffles, there at most 32 rising sequences.
But the reversed deck, numbered 52 down to 1, has 52 rising sequences! Therefore the reversed deck cannot be obtained in 5 random riffle shuffles!
So the maths behind this fact is the analysis of shuffling involves both combinatorics, probability, and even some group theory. Though we've been discussing "random" riffle shuffles in this Fun Fact, you can also study the mathematics of Perfect Shuffles, which have no randomness.
Huh.. so many explanation.
9. How many Prime?
Are there infinitely many primes?
I'll give a proof to show that there must be infinitely many primes. We will show that if there were only finitely many primes, it would lead to a contradiction.
First note that if two numbers differ by one, then they cannot have any common factors (or else that common factor would also divide 1).
So now suppose there are only finitely many primes p1, p2, ..., pn. Then by multiplying them all together, we get a (very large!) integer N that must be divisible by all the primes. But then (N+1) cannot be divisible by any prime, because N and N+1 have no common factors. This is clearly impossible because it contradicts the fact that every integer greater than 1 can be factored into primes.
For example, suppose 2, 3, and 5 were the only primes. Then (2)(3)(5)+1=31 cannot be divisible by any of them and must be divisible by some other prime number (in this case 31) which shows that 2, 3, 5 could not be the complete set of primes.
Note that this proof does not say that N+1 necessarily itself must be a prime... it just shows that it must be divisible by one that was not in the original list.
10. Ride a bike with Square Wheels
So, it's common that a bike's wheels is round, right?. But do you ever imaging riding with a square wheels? Are you crazy? It's impossible to ride with square wheels.
Don't worry, it's actually possible to ride with square wheels. But of course how this works is by changing the road itself into a curve like hill shape or cosh curve shape or also call as Catenary. A square wheel can roll smoothly if the curve is in a right shape. Imagine you riding this with this kind of bike. Pure awesome, sort off.
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