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TimMann
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Mark Twain didn't know
« on: Oct 14th, 2003, 1:17am » |
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Here's a funny old quote from Mark Twain.
If it would take a cannon ball 3 1/3 seconds to travel four miles, and 3 3/8 seconds to travel the next four, and 3 5/8 to travel the next four, and if its rate of progress continued to diminish in the same ratio, how long would it take to go fifteen hundred million miles?
--Arithmeticus, Virginia, Nevada
I don't know.
--Mark Twain
Despite the fact that MT was obviously lampooning folks like us who enjoy oddball math problems, does anyone want to seriously take a crack at this one?
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ThudnBlunder
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Re: Mark Twain didn't know
« Reply #1 on: Oct 14th, 2003, 2:40am » |
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Quote:| does anyone want to seriously take a crack at this one? |
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3 1/3 : 3 3/8 : 3 5/8 = 80 : 81 : 87
Sloane gives 10 sequences containing the sub-sequence 80, 81, 87
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wowbagger
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Re: Mark Twain didn't know
« Reply #2 on: Oct 14th, 2003, 5:12am » |
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on Oct 14th, 2003, 2:40am, THUDandBLUNDER wrote:| 3 1/3 : 3 3/8 : 3 5/8 = 80 : 81 : 87 |
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That's exatcly the problem. There is no "same ratio". Of course, the intention should be clear, but I don't feel like doing such a calculation right now.
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ThudnBlunder
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Re: Mark Twain didn't know
« Reply #3 on: Oct 14th, 2003, 11:29am » |
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Quote:| There is no "same ratio". Of course, the intention should be clear... |
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And so should your meaning. 
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wowbagger
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Re: Mark Twain didn't know
« Reply #4 on: Oct 15th, 2003, 7:48am » |
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on Oct 14th, 2003, 11:29am, THUDandBLUNDER wrote:| And so should your meaning. |
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I'm not sure what you mean. The riddle tries to imply that the nth four miles take the cannon ball 3+(2n-1)/8 seconds.
If my calculation is correct, it'd take the ball over 500 million years. If it's wrong, that's because I was discussing much more important things juts a few minutes ago. 
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| « Last Edit: Oct 15th, 2003, 7:49am by wowbagger » |
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jtrook
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Re: Mark Twain didn't know
« Reply #5 on: Oct 15th, 2003, 8:32am » |
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How do you determine a sequence based on three numbers with no perceptible pattern?
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towr
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Re: Mark Twain didn't know
« Reply #6 on: Oct 15th, 2003, 8:55am » |
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on Oct 15th, 2003, 8:32am, jtrook wrote:| How do you determine a sequence based on three numbers with no perceptible pattern? |
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Given any sequence, however short, you can allways find a 'pattern'. And if it's simple enough it may even be reasonable to assume that it's the real pattern behind a sequence..
of course 3+(2n-1)/8 seconds doesn't fit, with 3 1/3 for n=1
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aero_guy
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Re: Mark Twain didn't know
« Reply #7 on: Oct 15th, 2003, 11:01am » |
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This works: (5n2-3n+160)/48 when n starts from 0.
I changed the way it sums a little to help with the speed, here it is in matlab:
time=0;
for i=0:374999999
time=time+5*i^2-3*i;
end
time=(time+160*375000000)/48/60/60/24/365.2425
It is going to take a while to run. Is there some way of integrating the equation so we can get the answer explicitly?
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towr
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Re: Mark Twain didn't know
« Reply #8 on: Oct 15th, 2003, 11:18am » |
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summing (5n^2-3n+160)/48 from 0 to i gives (i+1)(5i^2 - 2i + 480)/144.
So the answer would be 5493163710937880468750/3
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aero_guy
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Re: Mark Twain didn't know
« Reply #9 on: Oct 15th, 2003, 11:30am » |
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I get 58.024 million billion years.
14.648 million billion years is spent in the last 4 miles. I think we are hitting quantum velocities there.
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ThudnBlunder
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Re: Mark Twain didn't know
« Reply #10 on: Oct 15th, 2003, 12:18pm » |
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Quote:And if it's simple enough it may even be reasonable to assume that it's the real pattern behind a sequence..
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Get real! There is no 'real' here. We simply have three numbers.
Everything else comes from your own imagination. And especially that of your souped-up computer. 
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| « Last Edit: Oct 16th, 2003, 12:50am by ThudnBlunder » |
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towr
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Re: Mark Twain didn't know
« Reply #11 on: Oct 15th, 2003, 2:12pm » |
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on Oct 15th, 2003, 12:18pm, THUDandBLUNDER wrote:
Get real! There is no 'real' here. We simply have three numbers.
Everything else comes from your own imagination. And especially that of your souped-up computer. |
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I never said there was a 'real' there, but there may be.. Someone may have posed those three numbers with a pattern in mind, and that would be the 'real' pattern behind the sequence, the one which determines how the creater would continue the sequence. Whether we can find it is another question.
And naturally even if there isn't a pattern we may well find one, or if there is a pattern we may instead declare it is chaos when we can't find it.. Those are simply the Erisian and an-Erisian principles..
And even reality itself may not exist, it may just be your imagination.. But let's assume it isn't, so we can discuss this further without assuming you're delirious 
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| « Last Edit: Oct 15th, 2003, 2:27pm by towr » |
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wowbagger
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Re: Mark Twain didn't know
« Reply #12 on: Oct 16th, 2003, 3:28am » |
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on Oct 15th, 2003, 8:55am, towr wrote:| of course 3+(2n-1)/8 seconds doesn't fit, with 3 1/3 for n=1 |
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My eager pattern recognition algorithm seems to have ovverridden careful reading.
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ThudnBlunder
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Re: Mark Twain didn't know
« Reply #13 on: Oct 16th, 2003, 8:33am » |
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Quote:| The riddle tries to imply that the nth four miles take the cannon ball 3+(2n-1)/8 seconds. |
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Tries? The riddle tries to imply nothing of the sort. You do.
You are implying that the data must fit a quadratic, whereas there are an infinite number of polynomials that would serve equally well.
Quote:| I never said there was a 'real' there, but there may be.. |
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Pure speculation, signifying nothing.
Quote:| And even reality itself may not exist, it may just be your imagination.. |
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Define 'reality'. (Don't use your imagination.)
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| « Last Edit: Oct 16th, 2003, 8:49am by ThudnBlunder » |
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towr
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Re: Mark Twain didn't know
« Reply #14 on: Oct 16th, 2003, 8:58am » |
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on Oct 16th, 2003, 8:33am, THUDandBLUNDER wrote:
Tries? The riddle tries to imply nothing of the sort. You do. |
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The person who created the puzzle might have been implying something as well.
Quote:| You are implying that the data must fit a quadratic, whereas there are an infinite number of polynomials that would serve equally well. |
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Occam's razor
Quote:| Pure speculation, signifying nothing. |
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It isn't speculation that it may be so, whether it is so is speculation.. The 'may' is specifically there to denote that it's speculation.. And it's not speculation that it's speculation.
Quote:| Define 'reality'. (Don't use your imagination.) |
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In this context it's what you think is real (like the world, the internet, me, other people, your keyboard, etc).
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ThudnBlunder
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Re: Mark Twain didn't know
« Reply #15 on: Oct 16th, 2003, 9:15am » |
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Everything I have to say about this puzzle is contained within my first post in this thread. When I make a (mathematical) statement I like to be as sure as possible that what I say is true. Speculation I leave for the physicists.
Quote:| In this context it's what you think is real (like the world, the internet, me, other people, your keyboard, etc). |
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You are confusing reality with perception. In this case, the reality is unknowable.
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| « Last Edit: Oct 16th, 2003, 9:33am by ThudnBlunder » |
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towr
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Re: Mark Twain didn't know
« Reply #16 on: Oct 16th, 2003, 9:23am » |
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on Oct 16th, 2003, 9:15am, THUDandBLUNDER wrote:| You are confusing reality with perception. |
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No, I'm not.. You're just perceiving it like that.. It depends on the context.
Our 'reality' is formed by our perception.. Just like virtual reality is formed by computers.. Those are examples of one interpretation of 'reality', with their own context.
Another interpretation is that of absolute objective truth. And that interpretation has another context.
Words have no intrinsic meaning, we give them meaning. 'reality' means nothing, except by our grace. And I'm sure you understood what I meant, and that was indeed it's meaning.
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wowbagger
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Re: Mark Twain didn't know
« Reply #17 on: Oct 16th, 2003, 9:27am » |
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on Oct 16th, 2003, 8:33am, THUDandBLUNDER wrote:
[quote 1]
Tries? The riddle tries to imply nothing of the sort. You do.
You are implying that the data must fit a quadratic, whereas there are an infinite number of polynomials that would serve equally well.
[quote 2]
Pure speculation, signifying nothing.
[quote 3]
Define 'reality'. (Don't use your imagination.) |
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T&B, would it be too much to ask to keep the header of the quote, so that one can distinguish quotes from different people? It's really a nuisance, and may well confuse readers, even those you quote.
I didn't imply the data must fit a quadratic - only if you view the linear dependence as a special case of a quadratic.
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ThudnBlunder
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Re: Mark Twain didn't know
« Reply #18 on: Oct 16th, 2003, 9:57am » |
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on Oct 16th, 2003, 9:27am, wowbagger wrote:
T&B, would it be too much to ask to keep the header of the quote, so that one can distinguish quotes from different people? It's really a nuisance, and may well confuse readers, even those you quote. |
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OK, if you insist. Personally, I think you should have been made an Ubermoderator long ago. 
on Oct 16th, 2003, 9:27am, wowbagger wrote:
I didn't imply the data must fit a quadratic - only if you view the linear dependence as a special case of a quadratic. |
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I am not sure what you mean here - a linear difference implies a quadratic function.
towr, I am concerned with the reality of the puzzle, not with our perception of it (which you call "our 'reality'"). With regards to this puzzle, as the former is unknowable, I can see no purpose that is served by replacing it with the latter, especially given the history of the puzzle as explained by Tim Mann.
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| « Last Edit: Oct 16th, 2003, 9:53pm by ThudnBlunder » |
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wowbagger
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Re: Mark Twain didn't know
« Reply #19 on: Oct 16th, 2003, 10:42am » |
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on Oct 16th, 2003, 9:57am, THUDandBLUNDER wrote:
I am not sure what you mean here - a linear difference implies a quadratic function. |
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Well, yeah, probably my fault once again. Even less reason for me to challenge your Überpuzzler status.
By the way, I think it's okay to have only a "Quote:" header if one splits a single post. That's what I do anyway.
Quote:| towr, I am concerned with the reality of the puzzle, not with our perception of it (which you call "our 'reality'"). With regards to this puzzle, as the former is unknowable, I can see no purpose that is served by replacing it with the latter, especially given the history of the puzzle as explained by Tim Mann. |
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You're right in that every interpretation by us is a guess at the original one (assuming there was one). On the other hand, for every finite sequence of numbers we could make up multiple rules that generate it. So in a way it's always about finding the one that was intended - and that one is, hopefully, the most plausible mathematically.
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| « Last Edit: Oct 16th, 2003, 10:43am by wowbagger » |
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towr
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Re: Mark Twain didn't know
« Reply #20 on: Oct 16th, 2003, 10:45am » |
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on Oct 16th, 2003, 9:57am, THUDandBLUNDER wrote:| towr, I am concerned with the reality of the puzzle, not with our perception of it (which you call "our 'reality'"). With regards to this puzzle, as the former is unknowable, I can see no purpose that is served by replacing it with the latter, especially given the history of the puzzle as explained by Tim Mann. |
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Euhm.. given that the reality isn't known, 'our reality' is the only thing we can use. If the former were known, then there would be truely no reason to use a substitution. But since it isn't known we do need a substitution, or leave the puzzle alone, which defeats the purpose of a puzzle..
Unless you create a puzzle yourself you can never know the reality, only your perception of it. Perception and our mind are the only things we have to discover the reality of 'our reality', as direct knowledge of it isn't possible afaik.
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TimMann
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Re: Mark Twain didn't know
« Reply #21 on: Oct 16th, 2003, 11:38am » |
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I'm really amused that I managed to set off such a long philosophical debate by posting this puzzle.
My own opinion is that Twain didn't intend for the puzzle to have a solution. Despite the mention of a physical object, there doesn't seem to be any sort of physical meaning that would tell us what kind of equation to fit the given numbers to. Fitting them to a quadratic is as good as any other idea, and is the simplest in some sense, but it isn't clearly "correct."
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| « Last Edit: Oct 16th, 2003, 11:39am by TimMann » |
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