BS: Who else plays little # games?

Subject: RE: BS: Who else plays little # games?
From: GUEST,Grishka
Date: 05 Sep 13 - 07:53 AM

Snail, you are the one who claims maths proficiency. n=5 works, the cases of n up to 4 can be tested manually or by a little computer script. My friend was not too specific in his email.

Little # games? Questions of the type "is this true for all integers?" are often easy to ask and difficult to answer with a proof. This one turned out to be of medium difficulty, others like Fermat's theorem took centuries to be answered by professional mathematical researchers. Football is a little ball game, but we might not enjoy playing it against Bayern München or Real Madrid if we seriously want to win.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 05 Sep 13 - 09:58 AM

("The details are left to the reader.")

There's a challenge.

Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned.

Sorry, not proved by your friends argument.

(I only claimed to have "a bit more than school maths".)

Subject: RE: BS: Who else plays little # games?
From: GUEST,Grishka
Date: 05 Sep 13 - 11:16 AM

Snail, I hope you are just trying to tease me. I'll join the game one more time, though I fear that readers are already bored.

   Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned.

Sorry, not proved by your friends argument.

As I wrote before, it was my error; n can in fact be either 1, or 2, or 3, or 4, by that argument. This leaves us with checking all values of x from 1 to 99 - a "finite" task.

   ("The details are left to the reader.")

There's a challenge.
(I only claimed to have "a bit more than school maths".)

It does not seem too difficult to me, just boring. Hint: if the inequality holds for some n (> 1), it will also hold for n+1, since the left side is nonnegative and increases by a factor 10, whereas the right side only increases by a factor (1 + 2/n + 1/n²), which is less than 4. By induction, the assertion is proved.

Wrong again? Requiring non-school maths?

Subject: RE: BS: Who else plays little # games?
From: GUEST,Grishka
Date: 05 Sep 13 - 11:25 AM

Instead of "nonnegative" read "positive" - pedants!

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 05 Sep 13 - 01:12 PM

Snail, I hope you are just trying to tease me.

Not at all. Just flexing my creaking mental muscles on an interesting problem. I thought that was what this thread was about. If you didn't want your friends "proof" discussed, why did you post it?

Hint: if the inequality holds for some n (> 1), it will also hold for n+1...

Yes, the assertion is true for all values of n greater than or equal to 5. No problem.

n can in fact be either 1, or 2, or 3, or 4, by that argument. This leaves us with checking all values of x from 1 to 99 - a "finite" task.

That falls short of your friends claim he had a proof of MtheGM's conjecture that 1 and 81 were the only numbers with this property. It does not rule out the possibility that there could be 3 and 4 digit numbers that share the porperty. OK, there aren't but that isn't the point.

Have you got back to your friend about this?

(All right, I might be teasing a bit.)

Subject: RE: BS: Who else plays little # games?
From: GUEST,Grishka
Date: 05 Sep 13 - 01:53 PM

You know the answer, Snail, and you have read that my friend left the details to me (which well he could; if this were serious rather than a "game", I would have been more cautious and meticulous).

Remains the question whether we enjoy such "games". I do to some degree, but only once in a while. This one will last for the rest of this year. In fact I must confess generally that a part of my personality enjoys discussing details, whereas the larger part strongly objects to the negligence of more important problems. I cannot really complain about lack of genuine challenges for my mental muscles; posting to Mudcat may amount to escapism altogether. One of my excuses is that it is a good training for my proficiency in English.

Some training in mathematics and logic has a great practical value as well: it helps us to be not so quickly convinced by apparent "common sense", neither by others nor by ourselves.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 08 Sep 13 - 04:08 PM

Grishka, nothing you have posted so far would have suggested that English wasn't your first language.

if this were serious rather than a "game", I would have been more cautious and meticulous).

Sorry, don't get that. Whether it's serious or a game, I can't see the point of doing it if you aren't going to do your best to get it right.

I'm sorry if we'v scared off the other readers but MtheGM must have known that he was setting a chalenge. If he doesn't like the answer, he has only himself to blame.

Maybe you or I should try and explain your friends proof in plain English. "10^n-1 ≤ x²" doesn't make a lot of immediate sense to someone who isn't used to mathematical notation.

Subject: RE: BS: Who else plays little # games?
From: MGM·Lion
Date: 08 Sep 13 - 04:20 PM

No, I didn't. Don't tell me what "I must have known", please. I asked if anyone else played these little games. For all Snail's thinking, three threads back, that these great big suge ginormous non-games are "what this thread is about"--

It's my thread.

And it isn't.

So yah-sucks-boo! to all you hijacking spoilsports & party-poopers.

Likewise Chiz-Chiz!

And great big botty-boo-bums!

ɷɷɷɷɷɷɷɷɷɷɷ

Subject: RE: BS: Who else plays little # games?
From: Nigel Parsons
Date: 08 Sep 13 - 08:57 PM

Snail:
Grishka, I would have thought that my method was more accessible to other readers than the epsilon stuff. I reproduce it here -
1/9 = 0.1111....
9 * 1/9 = 9 * 0.1111....
1 = 0.9999....
I may have been a bit sloppy with the notation for recurring. It could have been -
1/9 = 0.(1)
9 * 1/9 = 9 * 0.(1)
1 = 0.(9)
but I think people knew what I meant.
I'm still unclear as to what is specifically dodgy about it.
Your problem is in the first line: 1/9 = 0.1111....
(Assuming the multiple dots mean 'recurring')
NO 1/9 approximates to 0.111(recurring). No matter how many decimal places you add, you will never get a valid value of 1/9 in decimal notation.
1/9=3/27, or 4/36 etc.
To work with 1/9 mathematically, you do not replace it with its decimal approximation until you have calculated all other transactions. To do so is to bring in inaccuracies which may be multiplied by other transactions.

Subject: RE: BS: Who else plays little # games?
From: DMcG
Date: 09 Sep 13 - 02:47 AM

You are right, Nigel, but as I said earlier, to convert the method into a mathematical proof needs a few more powerful tools that would obscure the heart of idea, which is easy to grasp for a non-mathematician. This is essentially the same problem that crops up again and again in science and mathematics (and, indeed history and many other subjects): to teach something you often have to simplify it so much that you know is not strictly true. Alternatively, you can stick to truth (or I should say our best current understanding), and present something that is so convoluted and complex it is, quite literally, incomprehensible.

You may remember that there was a TV show a looooong time ago called 'That was the week that was'. The intro varied each week, but on one occasion it contained the couplet "One eye open wide, one eye closed. And between the two the picture gets composed'. I've often thought that is one of the wisest remarks I've ever heard from a television program. And it applies here.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 09 Sep 13 - 04:35 AM

DMcG has answered better than I could but, as I said, I think my "proof" works well enough in the context of little # games.

I'm a bit puzzled, though, by "until you have calculated all other transactions". What other transactions? The process of dividing by 9 leaving remainder 1 isn't going to change.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 09 Sep 13 - 05:00 AM

Come off it MtheGM. Do you seriuosly expect me to believe that you didn't realise that your question would unleash a hoard of geeks, nerds and anoraks? (Well, four or five of us.)

You asked the question. Stop trying to evade your responsibility. IT'S ALL YOUR FAULT.

Subject: RE: BS: Who else plays little # games?
From: GUEST,Grishka
Date: 09 Sep 13 - 05:04 AM

MichaelTheGM, may I remind you of your OP

Do any other squares do that? And does anyone else find such things of as much interest ...

You consider those who try to answer your questions "hijacking spoilsports"?

Subject: RE: BS: Who else plays little # games?
From: MGM·Lion
Date: 09 Sep 13 - 06:03 AM

Eeeeeeeeeeeeeeekkkk! I've opened Pandora's Box. Honest, Zeus, I never meant ~~~~~

Aaaaarrrggggghhhhh. It's off-down-the-garden-to-eat-worms-time again...

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 09 Sep 13 - 06:39 AM

Come now, Michael, don't underestimate yourself. As has been said, the maths involved is school level, the sort of stuff you were doing 65 years and more ago. The maths is just used to express ideas in a form that can then be manipulated mathematically. It could all be done in plain English but at considerably more length.

Go on, give it a try. You might find you enjoy it.

Subject: RE: BS: Who else plays little # games?
From: Nigel Parsons
Date: 16 Sep 13 - 12:52 PM

From: TheSnail - PM

I'm a bit puzzled, though, by "until you have calculated all other transactions". What other transactions? The process of dividing by 9 leaving remainder 1 isn't going to change.

The 'other transactions' are that you started by showing 1/9= 0.111..
(which is only an approximation) and then showed:

9 * 1/9 = 9 * 0.1111....
1 = 0.9999....

It is in this second stage that you have overcomplicated things by replacing 1/9 by a close approximation of 1/9.
9* 1/9 = 9/1 * 1/9 = 9/9 =1

All you have shown is that an approximation of one ninth is very close to the value of One.

Subject: RE: BS: Who else plays little # games?
From: TheSnail
Date: 19 Sep 13 - 11:44 AM

Maybe it's a matter of notation although I think O.111... is a recognised abreviation for 0.1 recurring. Would O.(1) be better? I don't think it's an approximation. Pi written out to a trillion places of decimals would be an approximation but 0.(1) doesn't mean 0.111... written out to a trillion places of decimals, it means written out to an infinite number of decimal places. Infinity doesn't just mean a very big number. As far as I can see, 1/9 is exactly equal to 0.(1). Just because you can't write it out in full doesn't mean it isn't true.

Others can argue this better than me - http://en.wikipedia.org/wiki/0.999%E2%80%A6