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

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

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

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.**

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

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

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.**

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

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

Hi Howardroark;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

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

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)

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

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

Hi gAr;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

Hi bobbym,

Thanks. Please check post #260 also.

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

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

Hi gAr;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

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

Hi bobbym,

Nice explanations in that thread, great!

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

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

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

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

Yes, and I never encountered these problems before.

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

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

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

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

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

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

Hi gAr;

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

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

Hi,

Ok, thanks for looking at it.

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

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

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

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

Hi bobbym,

Ok, no problem. I'll check.

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

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

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:

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

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

Hi gAr;

Can I see your 3x3?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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

Hi bobbym,

Here it is:

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

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

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?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**