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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,560

Even more "Stephen Froggatt" puzzles::

Pet Show

Newspaper Caper

Three Of The Best

A Brave Puzzle

A Weighty Problem

The Make-You-Very-Cross Number

NIM-ble Thinking

MatheMusic

A Hole New Board Game

Matches Of The Day

Ancestrally Speaking

Have You Herd Of Cows

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,914

hi

Sorry ... bit of a rush.

Weighty Puzz think I have another soln.

More later

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,675

Hi MIF;

Funny that Bob and I went for the same problem.

For the weighty problem there are 10 solutions to the problem. Not counting permutations. Want to add the rest? Hope I did not prune out any others.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,560

Great! I will put that in the solution.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,675

Hi MIF;

A mathematical solution would be nice but I do not have one.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

I would think there is one, thought it does seem GFs might be of use.

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,675

Hi anonimnystefy;

GF's usually count the number of solutions not the type of solutions. For that, I usually call on number theory.

Still in all a gf played an important part in the program. Without it there would be no program.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**Thurhame****Member**- Registered: 2013-03-03
- Posts: 6

There is a problem with the "Three Of The Best" puzzle.

The puzzle claims there are two solutions to "1+(1/(1+(1/...)))". However, it actually converges to the golden ratio

MORE COMPLEX PROOF:

The formula is the limit as n goes to infinity of the sequence

Assume some exists (otherwise the sequence would be undefined).

Define a sequence .

We proceed by induction.

Base step:

Inductive Hypothesis:

Thus we have proved by induction the relation for all .

Since is a sequence of the form , this ratio converges to the golden ratio.

*Last edited by Thurhame (2013-03-05 13:28:02)*

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Wonderful, Thanks!

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

Hi Thurhame

Can you specify where it was claimed that 1+1/(1+1/(1+...)) is equal to 2?

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

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,675

Hi;

The answer given is 2 when it should be the golden ratio.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**Thurhame****Member**- Registered: 2013-03-03
- Posts: 6

anonimnystefy wrote:

Hi Thurhame

Can you specify where it was claimed that 1+1/(1+1/(1+...)) is equal to 2?

Oh, my mistake, it just said there were two solutions to the formula. However, that's still wrong; my proof shows that any solution must be equal to the golden ratio, i.e. there are at most 1 solutions.

Edited my post above. Thanks!

*Last edited by Thurhame (2013-03-04 05:26:01)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

Hi Thurhame

Yes, with that I agree. The answer should be one. I think they didn't consider that the number must be positive.

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

The knowledge of some things as a function of age is a delta function.

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**Thurhame****Member**- Registered: 2013-03-03
- Posts: 6

Ah, thanks for reminding me, my simple less-rigorous proof isn't adequate if i'm only proving the number of solutions, rather than the value. Removing it now. Hope the more complex proof doesn't make anyone's eyes glaze over.

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