quick tricks

[darksaber21]

Don't worry, I was bad on the approximations at first as well. But now I can do them pretty darn well, with some blimps here and there. Unfortunately, it's not saying much as I average reaching number 50 for every test....

But, I know how to do all the problems, if that counts, haha...

[PokémonMaster]

[quote="darksaber21]But, I know how to do all the problems, if that counts, haha...[/quote]

It coumts for a lot, actually. If you know everything on the test, it is kust a matter of gaining speed as opposed to needing to gain speed and learn more things.

[101dalmatians]

[qwerty123]

[101dalmatians]

[101dalmatians]

So 1 dry quart is 1/32 a bushel...

EDIT...

[qwerty123]

[101dalmatians]

[qwerty123]

I know that division by 37 is 3 times the repeating digit, but what is the trick for division by 27? Such as...
[unparseable or potentially dangerous latex formula]________________ (mixed #)

The sum of the squares of the regular fib sequence is "last * next term", but what is it for any fibonacci-type sequence?
Ex. [unparseable or potentially dangerous latex formula]

is there a trick for slant asymtotes besides long div and/or synthetic div?

and to darksaber...
was it your birthday a couple of days ago? (saw on bottom of forum)...
just wondering
happy late birthday!

[darksaber21]

[darksaber21]

[101dalmatians]

[101dalmatians]

OMG! Im totally tired of this!
I see this so frequently that my avg score has gone down - b/c of this.
Ugh.

Ok:

Like this:

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

I know how to do this, but Im not able to do this quickly - like in 10 sec. How do you do it fast?

[8055]

What are happy numbers? Others like this? I get stuck on these questions.

[101dalmatians]

Happy numberFrom Wikipedia, the free encyclopediaJump to: navigation, search Not to be confused with Harshad number (derived from Sanskrit harsa: "great joy").
A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers[1]).

Contents [hide]
1 Overview
2 Sequence behavior
3 Happy primes
4 Special happy numbers
4.1 Happy pythagorean triplets
5 Happy numbers in other bases
6 Cubing the digits rather than squaring
7 Origin
8 Popular culture
9 Programming example
10 References
10.1 Additional resources
11 External links


[edit] OverviewMore formally, given a number , define a sequence , , ... where is the sum of the squares of the digits of . Then n is happy if and only if there exists i such that .

If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of the sequence are unhappy.

For example, 19 is happy, as the associated sequence is:

12 + 92 = 82
82 + 22 = 68
62 + 82 = 100
12 + 02 + 02 = 1.
The happy numbers below 500 are:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496 (sequence A007770 in OEIS).
The happiness of a number is preserved by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number.

The unique combinations of the happy numbers below 1,000 follow (the rest are just rearrangements and/or insertions of zero digits):

1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899.(sequence A124095 in OEIS).
[edit] Sequence behaviorIf n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle

4, 16, 37, 58, 89, 145, 42, 20, 4, ...
To see this fact, first note that if n has m digits, then the sum of the squares of its digits is at most , or .

For and above,


so any number over 1000 gets smaller under this process and in particular becomes a number with strictly fewer digits. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243.

In the range 100 to 243, the number 199 produces the largest next value, of 163.
In the range 100 to 163, the number 159 produces the largest next value, of 107.
In the range 100 to 107, the number 107 produces the largest next value, of 50.
Considering more precisely the intervals [244,999], [164,243], [108,163] and [100,107], we see that every number above 99 gets strictly smaller under this process. Thus, no matter what number we start with, we eventually drop below 100. An exhaustive search then shows that every number in the interval [1,99] either is happy or goes to the above cycle.[citation needed]

The above work produces the interesting result that no positive integer other than 1 is the sum of the squares of its own digits, since any such number would be a fixed point of the described process.

There are infinitely many happy numbers and infinitely many unhappy numbers. As an informal proof, consider the following:

For any happy number, every number in its sequence is happy. Likewise for unhappy numbers.
For this reason, to come up with a new happy number, it is sufficient to find a number such that the sum of the squares of its digits is itself a happy number. Likewise, to find a new unhappy number, one can simply find a number such that the sum of the squares of its digits leads to an unhappy number.
It is easy to generate a number that leads (in this sequence) to a known happy or unhappy number. For example, if one wonders if any number will produce 14,308, say, the quick response is to write down the digit "1" 14,308 times and you have created such a number.
Infinitely many such numbers exist, since zeros can be inserted at will.
The first pair of consecutive happy numbers is 31, 32.[2] The first set of triplets is 1880, 1881, and 1882.[3]

An interesting question is to wonder about the density of happy numbers. In the interval , 15.5% (to three significant figures) are happy.(sequence A068571 in OEIS)

[edit] Happy primesA happy prime is a number that is both happy and prime. The happy primes below 500 are

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence A035497 in OEIS).

All numbers, and therefore all primes, of the form 10n + 3 and 10n + 9 for n greater than 0 are Happy (This does not mean that these are the only happy primes, as evidenced by the sequence above). To see this, note that

All such numbers will have at least 2 digits;
The first digit will always be 1 due to the 10n
The last digit will always be either 3 or 9.
Any other digits will always be 0 (and therefore will not contribute to the sum of squares of the digits).
The sequence for adding 3 is: 12 + 32 = 10 → 12 = 1
The sequence for adding 9 is: 12 + 92 = 82 → 82 + 22 = 64 + 4 = 68 → 62 + 82 = 36 + 64 = 100 -> 1
The palindromic prime 10150006 + 7426247×1075,000 + 1 is also a happy prime with 150,007 digits because the many 0's do not contribute to the sum of squared digits, and , which is a happy number. Paul Jobling discovered the prime in 2005.[4]

As of 2010[update], the largest known happy prime is (Mersenne prime). Its decimal expansion has 12,837,064 digits.[5]

[edit] Special happy numbers986,543,210 : Greatest happy number with no repeated digits[citation needed]
1,234,456,789 : Smallest zeroless pandigital happy number[citation needed]
10,234,456,789 : Smallest pandigital happy number[citation needed]
13,456,789,298,765,431 : Smallest zeroless pandigital palindromic happy number[citation needed]
1,034,567,892,987,654,301 : Smallest pandigital palindromic happy number[citation needed]
[edit] Happy pythagorean tripletsAll Pythagorean triplets with all integers happy and less than 10000[citation needed]
( 700, 3465, 3535) ( 748, 8211, 8245) ( 910, 8256, 8306) ( 940, 2109, 2309)
( 940, 4653, 4747) (1092, 1881, 2175) (1323, 4536, 4725) (1527, 2036, 2545)
(1785, 3392, 3833) (1900, 1995, 2755) (1995, 4788, 5187) (2715, 3620, 4525)
(2751, 8360, 8801) (2784, 6440, 7016) (3132, 7245, 7893) (3135, 7524, 8151)
(3290, 7896, 8554) (3367, 3456, 4825) (3680, 5313, 6463) (4284, 5313, 6825)
(4633, 5544, 7225) (5178, 6904, 8630) (5286, 7048, 8810) (5445, 6308, 8333)
(5712, 7084, 9100) (6528, 7480, 9928)

[edit] Happy numbers in other basesThe definition of happy numbers depends on the decimal (i.e., base 10) representation of the numbers. The definition can be extended to other bases.

To represent numbers in other bases, we may use a subscript to the right to indicate the base. For instance, represents the number 4, and


Then, it is easy to see that there are happy numbers in every base. For instance, the numbers


are all happy, for any base b.

By a similar argument to the one above for decimal happy numbers, unhappy numbers in base b lead to cycles of numbers less than . If , then the sum of the squares of the base-b digits of n is less than or equal to


which can be shown to be less than . This shows that once the sequence reaches a number less than , it stays below , and hence must cycle or reach 1.

In base 2, all numbers are happy. All binary numbers larger than 10002 decay into a value equal to or less than 10002, and all such values are happy: The following four sequences contain all numbers less than :





Since all sequences end in 1, we conclude that all numbers are happy in base 2. This makes base 2 a happy base.

The only known happy bases are 2 and 4. There are no others less than 500,000,000.[6]

[edit] Cubing the digits rather than squaringAn interesting extension to the Happy Numbers problem is to find the sum of the cubes of the digits rather than the sum of the squares of the digits. For example, working in base 10, 1579 is happy, since:[citation needed]

13+53+73+93=1+125+343+729=1198
13+13+93+83=1+1+729+512=1243
13+23+43+33=1+8+64+27=100
13+03+03=1
In the same way that when summing the squares of the digits (and working in base 10) each number above 243(=3*81) produces a number which is strictly smaller, when summing the cubes of the digits each number above 2916(=4*729) produces a number which is strictly smaller.

By conducting an exhaustive search of [1,2916] one finds that for summing the cubes of digits base 10 there are happy numbers and eight different types of unhappy number:

those that eventually reach which perpetually produces itself.

those that eventually reach which perpetually produces itself.

those that eventually reach the loop

those that eventually reach which perpetually produces itself.

those that eventually reach the loop

those that eventually reach which perpetually produces itself.

those that eventually reach the loop

those that eventually reach the loop

Starting with the happy numbers and then following with the unhappy numbers in the order given above, the density of each type of number in the interval [1,2916] is 1.54%, 28.4%, 34.7%, 5.73%, 17.4%, 4.60%, 3.60%, 2.67% and 1.34% (all to 3 significant figures).

Intriguingly, the second type of unhappy number includes all multiples of three . This fact can be proved by the exhaustive search up to 2916 and noting that a number is a multiple of three if and only if the sum of digits is a multiple of three if and only if the sum of its cubed digits are a multiple of three. By similar reasoning, all happy numbers of this type must have a remainder of 1 when dividing by 3.

One interesting result which comes from the above work is that the only positive whole numbers which are the sum of the cubes of their digits are 1, 153, 370, 371 and 407.

[edit] OriginThe origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and Senior Lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" (Guy 2004:§E34).

[edit] Popular cultureIn the 2007 Doctor Who episode "42", a sequence of happy primes (313, 331, 367, 379) is used as a code for unlocking a sealed door on a spaceship about to collide with a sun. When the Doctor learns that nobody on the spaceship besides himself has heard of happy numbers, he asks, "Don't they teach recreational mathematics anymore?"

The contestants in the 2012 University Challenge final were asked to identify a sequence of numbers as happy primes in a picture round.

[edit] Programming exampleThe examples below apply the 'happy' process described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number. Everything else is book-keeping (for example, the Python example precomputes the squares of all 10 digits).

Simple test in Python to check if a number is happy.[7]
SQUARE = dict([(c, int(c)**2) for c in "0123456789"])
def is_happy(n):
s = set()
while (n > 1) and (n not in s):
s.add(n)
n = sum(SQUARE[d] for d in str(n))
return n == 1
[edit] References1.^ "Sad Number". Wolfram Research, Inc.. http://mathworld.wolfram.com/SadNumber.html. Retrieved 2009-09-16.
2.^ "Lower of pair of consecutive happy numbers". # Encyclopedia of Integer Sequences. http://oeis.org/A035502. Retrieved 8 April 2011.
3.^ "First of triples of consecutive happy numbers". # Encyclopedia of Integer Sequences. http://oeis.org/A072494. Retrieved 8 April 2011.
4.^ The Prime Database: 10^150006+7426247*10^75000+1
5.^ Prime Pages entry for 242643801 - 1
6.^ "Sloane's A161872 : Smallest unhappy number in base n", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7.^ Happy Number Rosetta Code
[edit] Additional resourcesWalter Schneider, Mathews: Happy Numbers.
Weisstein, Eric W., "Happy Number" from MathWorld.
Happy Numbers at The Math Forum.
Numberphile
Guy, Richard (2004). Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. ISBN 0-387-20860-7
[edit] External linksReg Allenby page
Retrieved from "http://en.wikipedia.org/w/index.php?title=Happy_number&oldid=531078854"
Categories: Base-dependent integer sequencesRecreational mathematicsHidden categories: All articles with unsourced statementsArticles with unsourced statements from March 2011Articles containing potentially dated statements from 2010All articles containing potentially dated statementsArticles with unsourced statements from September 2011Articles with unsourced statements from October 2011Articles with inconsistent citation formatsUse dmy dates from June 2011Navigation menuPersonal tools
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For more information go to

[101dalmatians]

BTW.If you want more info, look it up in google or PM me.

[darksaber21]

[101dalmatians]

Thanks :)

And BTW bout that long post - sxk1693 gave me the idea.
At first, I was about to post normally but u know - he gave me an idea - and I just tried it out.
Apparently it worked!
Anyway - what is the word limit for a post - anyone know?

[sxk1693]

What!??! This is called framing.

[101dalmatians]

[8055]

Now I became an "expert" in like going to the second page... It took me a long time to figure out.
Also, I really think there is no need to fight about whose idea this stuff was but I get what happy numbers are.
Thanks for the info :D

[101dalmatians]

[101dalmatians]

[sxk1693]

[101dalmatians]

Oh... Well I didn't know that