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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
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Find the number with the clues given. You may need to use some math on the way. I start with a problem, and then when people answers, then I make another problem.

Good luck!

I.

My number is:

1) A prime number.

2) Half of ten.

What number is it?

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

Hi

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
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Wow, anonimnystefy!

:):):):)

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

How about the largest number less than 100 that can only be written one way as a difference of

squares?

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,730

Hi;

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

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Yep! Privy to a bit of number theory, eh? Would make Fermat smile in his grave.

How about a base (other than two) for which one can write any positive number in terms of using no coefficients greater than one?

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

Hi noelevans,

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
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Whoops, bobbym! 97 is a prime number, but is 10/2=97?

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Hi phrontister!

That's using your noodle and your computer! Turns out to be a great combination for lots of problem

solving. I bet Gauss, Newton, Euler, etc. would have given a lot to have the computational power

that we have today. They might not have left us any math to discover!

One of the characteristics of primes larger than two is that there is only one way to write them as

a difference of squares. Composites always have at least two ways to do it. Each factorization of

a number greater than two corresponds to a difference of squares factorization. For example

15=1x15=8^2-7^2 and 15=3x5=4^2-1^2. 21=1x21=11^2-10^2 and 21=3x7=5^2-2^2.

So a number like 27 which has 2 factorizations has 2 ways to write it as a difference of squares.

Primes and perfect squares are neat, multifaceted numbers.

Have a blessed evening!

Edit: Oops! Limit the above discussion to odd non-square composite numbers. Even numbers and

odd perfect squares may have fewer difference of square factorizations, especially if we do

not allow zero as one of the numbers. Example 4=4x1=(5/2 + 3/2)(5/2 - 3/2) but these are

not integral. Also 4=2x2=(2+0)(2-0) won't do if we disallow zero.

For 81=9x9 we have no difference of square factorization if we disallow zero.

But 81=1x81=(41+40)(41-40) and 81=3x27=(15+12)(15-12).

for 9=1x9=(5+4)(5-4) but 9=3x3=(3+0)(3-0) won't work if zero is disallowed.

So the square of a prime has only one difference of squares factorization IF we

disallow zero.

*Last edited by noelevans (2012-12-07 15:59:47)*

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,730

Hi;

How about a base (other than two) for which one can write any positive number in terms of using no coefficients greater than one?

There is the Zeckendorf numbers which use Fibonacci numbers as the base. The coefficients do not have to be other than 1 and 0. For instance:

Hi julianthemath;

Whoops, bobbym! 97 is a prime number, but is 10/2=97?

Your problem's answer is 5 , noelevans' problem has the answer of 97.

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

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Zeckendorf representations --- VERY INTERESTING! Thanks for pointing them out.

Actually I was just thinking of a base as a single integer >1. I've never seen a SET being a base.

Wiki points out that this property is characteristic of the Fibonacci numbers so no other set of

numbers has this property ascribed to the Fibonacci numbers.

My question is a bit tricky. Any positive integral base greater than 1 could satisfy the conditions I

mentioned. The coefficients could be negative integers (introduced by a many named Colson in

- --

1729) for example 481 = 1x1000 - 5x100 - 1x10 -9x1 = 1519 where the bars above the digits

mean subtract instead of adding. So 481 is written without any coefficients >1.

But if we require that the absolute value of the integral coefficients not be greater than 1 then

base 2 and one other base will satisfy this.

I had this message written earlier and I previewed it but forgot to submit it, so I lost it.

Live and learn!

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

Hi noelevans,

Edit: Oops!

Glad I spotted that you'd spotted that. I was close to putting pen to paper about my failure in getting the factorization & difference of squares feature to work with 22 and 72, and was looking at testing some other evens and then more odds.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
- Website

Got it, bobbym!

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
- Website

II.

My number is:

1) A 3-digit number.

2) A square number.

3) A composite number.

4) Divisible by 11.

What is it?

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,730

Hi;

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

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

Hi julianthemath,

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
- Website

Okay!

Two official answers are:

121

484

Bobbym, Phrontister, you are correct!

bobbym wrote:

484

phrontister wrote:

121

:):):):):):):):):):):):):):):):):):)

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**mathgogocart****Member**- Registered: 2012-04-29
- Posts: 1,426

121 smilies,right.....LOL

Hey.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,730

Hi;

Bobbym, Phrontister, you are correct!

I like phrontister's answer better. Where that other gets his answer from I can only speculate.

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

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

bobbym wrote:

Where that other gets his answer from I can only speculate.

9 darts into a dartboard?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,730

Maybe, but he probably would miss the board!

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

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,845
- Website

III.

My number is:

1) A 5-digit number.

2) Is ranged from 100^2-109^2.

3) Has a "25" at the end.

What is it?

Tell me the **square number and the square root.**

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

I meant thump them in with a hammer...not throw them.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

Hi julianthemath,

*Last edited by phrontister (2012-12-08 11:17:09)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,730

Hi;

I meant thump them in with a hammer...not throw them.

That would be more accurate.

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

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**