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#251 2011-02-18 17:58:36

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

New problem!

How many positive integers <100 have the property that the sum of the digits plus the product of the digits equals the number?

A says) 10.
B says) 9
C says) You missed one B!
D says) He missed 2.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#252 2011-02-18 19:43:11

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#253 2011-02-18 20:33:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#254 2011-02-22 05:49:11

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,

HI;

Medium hard:

Three cards are dealt off of a well shuffled ordinary deck of playing cards.

What are the chances that the first card is a 5 and then the second card is a diamond and the third card is a 4?

A says) 1 / 991.

B says) That's wrong it is 1 / 663. Because:

B was going to explain more when...

C said) Forget that gibberish. A is right,it is 1 / 991.

Who is right?




There are 15 players on a team roster. I need to select 10 of them for the team, 3 of whom will be co-captains. How many ways can I select my 10 man team with it's 3 co-captains.

A says) 120, I counted them!

B says) think of the team as c,c,c,p,p,p,p,p,p,p,p,p,p,p,p and your pick is:

c,c,c,p,p,p,p,p,p,p so:

10 ! / ( 3! * 7! ) = 120 so A is right.

C says) Oh boy, both of you are wrong it's:

Who is right!

(post #110)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#255 2011-02-22 07:40:53

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#256 2011-02-23 05:10:25

Howardroark
Member
Registered: 2010-10-10
Posts: 382

Re: Is this cool with you?

Hi bobbym,

bobbym wrote:

New problem!

How many positive integers <100 have the property that the sum of the digits plus the product of the digits equals the number?

A says) 10.
B says) 9
C says) You missed one B!
D says) He missed 2.


I come from a civilization which has given the world the number 0..

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#257 2011-02-23 08:24:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi Howardroark;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#258 2011-02-24 02:13:37

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi;

Is there any easy way to compute the unit digit of the nth triangular number?

A) Yes there is, and I know it.

B) That is not possible you just have to put the number into (n(n+1)) / 2 and do the arithmetic.

C) Yes, use a computer.



A really tough one, only I and 2 other people in the world ( and one of them is an alien ) can get this one.

A believes that it will take an average of 36 throws of a dice to get a (1,1). He bets B that he is right. They roll a dice all day long and get an average that is close to 40 throws so B wins the money. A says B got incredibly lucky or the dice is loaded. B says A is an idiot because the expected number of throws is 42 not 36. Who is right and why?


Hard:

A continues the argument from the last problem. He is really mad that B got his money. B being a magnanimous sort of guy agrees to give A a chance to get his dough back. B says let's play another game.

We both throw a die at the same time. I will take (1,2) and you can have (1,1) we keep playing until one of us gets the two rolls in a row that we need. A says that is ridiculous, it is an even game. B says let's play anyway. A says okay. Is B hustling A or is it an even game?

(post #109)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#259 2011-02-24 05:47:29

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi;

What is the coefficient of x^2000 in :

A says 3^1000 + 2

B says 2^2000 + 2

C says 2^1000

D says You can't do it. It is too tedious.



Medium:

3 people draw 3 different random numbers. They notice that when they take 2 of them, call them x and y and plug them into:

it is divisible by thirty. They pick 3 more different random integers, same thing.

A) says everytime you pick 3 random positive integers there will always be 2 so that f(x,y) is divisible by thirty.

B) says it is just a coincidence. True, there are no small numbers where that doesn't hold but it will eventually fail.

C) says B is right. Here is a countereample (18735625374653746532611112342965142999^3 -1, 872653419864765342638499993333333^18, 4653625436475888888888888111111113773^25 )

Original work please, I already know the 2 methods shown to me. So I will be looking for a new approach.

(post #111)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#260 2011-02-24 10:53:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#261 2011-02-24 14:20:32

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,

Thanks. Please check post #260 also.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#262 2011-02-24 15:51:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#263 2011-02-24 16:01:35

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,

Nice explanations in that thread, great!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#264 2011-02-24 16:14:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

After I posed my question set here, that thread opened up totally unaware of what had been asked here. It is a really difficult point in probability.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#265 2011-02-24 16:28:25

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Yes, and I never encountered these problems before.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#266 2011-02-24 16:55:24

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

First time I saw any of the problems I posed here, I freaked out. Took me years to solve some of them. That is why I believe in an understanding of probability and combinatorics through generating functions, recurrences, Markov chains, distributions, series, renewal theory, trees, block walks etc. It is true they are clumsy on small problems but when you see a toughie you will have plenty of weapons.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#267 2011-02-24 17:28:49

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

I'm wondering what went wrong with analytical method for the sequence of (1,2) .


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#268 2011-02-24 18:40:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;

I do not know yet, I am looking at it now.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#269 2011-02-24 18:45:48

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi,

Ok, thanks for looking at it.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#270 2011-02-24 20:42:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;

I know by at least two other methods that the answer is 36 but I cannot make that expectation formula demonstrate that. I am wondering if the case of 1,1 is correct.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#271 2011-02-24 20:58:01

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,

Ok, no problem. I'll check.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#272 2011-02-25 05:27:48

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,

In post #13 ( http://www.mathisfunforum.com/viewtopic.php?id=14480&p=1 ),
You chose 4 states while obtaining (3,3). Can't we do it with 3 states just like what you did for obtaining (3,4)?

I get this:


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#273 2011-02-25 16:31:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;

Can I see your 3x3?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#274 2011-02-25 16:39:06

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Is this cool with you?

Hi bobbym,

Here it is:


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#275 2011-02-25 16:48:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,228

Re: Is this cool with you?

Hi gAr;

That is the one I am getting now. You are right in can be shortened to a 3x3. It seems there are lots of different ones you can come up with. Do you know of a way to always or almost always get the shortest one?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

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