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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,618

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

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,588

hi

Sorry ... bit of a rush.

Weighty Puzz think I have another soln.

More later

Bob

Children are not defined by school ...........The Fonz

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: 104,363

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: 7,618

Great! I will put that in the solution.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,363

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: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

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

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: 104,363

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!

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

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

I'm not crazy, my mother had me tested.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

Hi Thurhame

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

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: 104,363

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: Harlan's World
- Registered: 2011-05-23
- Posts: 16,015

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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