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Author Topic: Numerical Zendo  (Read 6321 times)

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WalrusMcFishSr

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Re: Numerical Zendo
« Reply #25 on: November 29, 2013, 09:12:28 pm »

Just double checked, I think that is correct.

ghostofmars, it doesn't have anything directly to do with prime numbers that I know of.

Here are a few clues if you want 'em:

5 is the lowest value in the sequence, you can't extrapolate it backwards any further
Arguably 65, 85, and 145 could be listed twice
Probably only the first three or four entries will be familiar to most
Just adding fours and eights (and now twelves) in convoluted ways won't do it...down the road there are larger differences between some successive numbers

More clues to follow if this stagnates.
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Re: Numerical Zendo
« Reply #26 on: November 30, 2013, 11:22:24 am »

Are the differences always multiples of 4?

Edit: Hey asking things like this makes it more like Numerical Zendo!
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

WalrusMcFishSr

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Re: Numerical Zendo
« Reply #27 on: November 30, 2013, 11:35:36 am »

I believe so. At least through the first 100 terms. I wonder if I could prove it to be generally true.

Hint: This sequence is most directly inspired by geometry.
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Re: Numerical Zendo
« Reply #28 on: November 30, 2013, 11:47:50 am »

Aha!

149, 157, 169, 173, 181

Explanation: The hypotenuse in Pythagorean Triples. Some numbers are arguably repeated because they can be formed from two different triangles, e.g. 162 + 632 = 332 + 562 = 652

Edit: I included a link but links are visible through spoiler tags. You can google Pythagorean Triples if you are curious though.
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

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Re: Numerical Zendo
« Reply #29 on: November 30, 2013, 11:52:34 am »

If the above is correct (which I'm reasonably certain it is) then Ghostofmars can go next, given I kind of sniped his first answer.
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

WalrusMcFishSr

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Re: Numerical Zendo
« Reply #30 on: November 30, 2013, 11:55:51 am »

Nice! Yep that's it.
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ghostofmars

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Re: Numerical Zendo
« Reply #31 on: November 30, 2013, 01:09:43 pm »

Let's try this
1, 6, 9, 10, 19, 26, 31, 44, 65, 74, 91, ...

It's perhaps a bit harder, so asking questions is encouraged :)
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heron

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Re: Numerical Zendo
« Reply #32 on: November 30, 2013, 01:13:33 pm »

Hmmm... are you sure that 65 isn't a 55?
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Re: Numerical Zendo
« Reply #33 on: November 30, 2013, 02:25:34 pm »

Observation: The difference between successive terms is odd
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

qmech

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Re: Numerical Zendo
« Reply #34 on: November 30, 2013, 03:17:03 pm »

Hmmm... are you sure that 65 isn't a 55?

I am also wondering this.
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ghostofmars

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Re: Numerical Zendo
« Reply #35 on: November 30, 2013, 05:52:56 pm »

Hmmm... are you sure that 65 isn't a 55?
Yes, 65 is correct.
Observation: The difference between successive terms is odd
That is true for the whole series.

I've extended the series a bit:
1, 6, 9, 10, 19, 26, 31, 44, 65, 74, 91, 116, 139, 160, 189, 216, ...
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ghostofmars

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Re: Numerical Zendo
« Reply #36 on: December 02, 2013, 03:23:27 pm »

A few more numbers
1, 6, 9, 10, 19, 26, 31, 44, 65, 74, 91, 116, 139, 160, 189, 216, 241, 274, 305, 334, 371, 406, 449, 490, 539, 576, ...

and the first tip
- can you find periodic features?
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qmech

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Re: Numerical Zendo
« Reply #37 on: December 02, 2013, 03:58:29 pm »

First differences: [5,3,1,9,7,5,13,21,9,17,25,23,21,29,27,25,33,31,29,37,35,43,41,49,37]
Second differences: [-2,-2,8,-2,-2,8,8,-12,8,8,-2,-2,8,-2,-2,8,-2,-2,8,-2,8,-2,8,-12]
Third differences: [0,10,-10,0,10,0,-20,20,0,-10,0,10,-10,0,10,-10,0,10,-10,10,-10,10,-20]

Nothing that lets me guess a term yet.
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ghostofmars

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Re: Numerical Zendo
« Reply #38 on: December 04, 2013, 04:34:00 pm »

1, 6, 9, 10, 19, 26, 31, 44, 65, 74, 91, 116, 139, 160, 189, 216, 241, 274, 305, 334, 371, 406, 449, 490, 539, 576, 621, 674, 725, 774, 821, 886, 949, 1000, ...

2nd tip: Plot the numbers or the differences.
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scott_pilgrim

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Re: Numerical Zendo
« Reply #39 on: December 04, 2013, 05:11:50 pm »



I can't figure anything out from it.  The one on the left is the original numbers and the one on the right is the differences.
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Re: Numerical Zendo
« Reply #40 on: December 04, 2013, 07:02:13 pm »

Judging by the (psuedo)derivative the function appears to be roughly quadratic. Perhaps see what the best approximation to a quadratic equation is and see if that gives any rough hints?
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

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Re: Numerical Zendo
« Reply #41 on: December 04, 2013, 07:15:24 pm »

Did it myself. Excel's best approximation is (roughly) 0.91x2 - 1.4x + 1.8. Comparing that to the data... I don't see anything of relevance.

Can I get a quick check: Are these numbers correct: ...774, 821, 886, ... ?
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

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Re: Numerical Zendo
« Reply #42 on: December 04, 2013, 07:19:40 pm »

Okay well I had a theory but it isn't panning out, so I'll just throw my observation/question out to the masses:

The second derivative is always of the form 10n - 2 for integer n - i.e. -12, -2, 8, 18 etc. Is this always the case?
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

heron

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Re: Numerical Zendo
« Reply #43 on: December 04, 2013, 07:56:06 pm »

...1069, 1136, 1201, 1274?
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ghostofmars

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Re: Numerical Zendo
« Reply #44 on: December 04, 2013, 08:10:53 pm »

Judging by the (psuedo)derivative the function appears to be roughly quadratic. Perhaps see what the best approximation to a quadratic equation is and see if that gives any rough hints?
Yes, it's quadratic +?. And the +? is the difficult part.

The second derivative is always of the form 10n - 2 for integer n - i.e. -12, -2, 8, 18 etc. Is this always the case?
Yes.

Can I get a quick check: Are these numbers correct: ...774, 821, 886, ... ?
I generate the numbers automatically. I can share the sheet. I hope, I set the privacy settings correctly ;)

...1069, 1136, 1201, 1274?
No, 1069, 1126, 1181, 1254
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heron

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Re: Numerical Zendo
« Reply #45 on: December 04, 2013, 08:19:07 pm »

Well, as I am prone to do, I cheated and found the answer in the google doc. The secret is to make a copy.
Anyway, it was pretty complicated.
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ghostofmars

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Re: Numerical Zendo
« Reply #46 on: December 04, 2013, 08:22:41 pm »

Well, as I am prone to do, I cheated and found the answer in the google doc. The secret is to make a copy.
Anyway, it was pretty complicated.
Actually, the generation in the spreadsheet, is more complicated than the generation by hand.
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Re: Numerical Zendo
« Reply #47 on: December 04, 2013, 08:39:16 pm »

It's very easy to cheat with google docs based information sharing. The general unwritten rule in PBF games is don't :).

Hmm...

I've added all the cells up to 100. You initially gave 11 numbers. Do you think that's enough to spot the 'complex' bit that's added completely?

Looks like the Quadratic without the extra bit, adding in those new terms, is most closely approximated by about 0.9x2 - 1.2x + 0.85. Is this correct?

Further looking, the period of the extra term seems to be 50. Also correct?

I feel like I've deduced (almost) everything I need now and just need to piece it together correctly to solve this now...
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...spin-offs are still better for all of the previously cited reasons.
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Re: Numerical Zendo
« Reply #48 on: December 04, 2013, 08:43:22 pm »

Further observation: Whatever the extra term on top of that 50 period cycle, can also be simplified to a 25 period cycle that reverses at the end. The extra term from the 9th to the 34th is the same as the one from the 34th to 59th but in the opposite direction (so t(33)=t(35), t(32)=t(36) etc.)
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...spin-offs are still better for all of the previously cited reasons.
But not strictly better, because the spinoff can have a different cost than the expansion.

ghostofmars

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Re: Numerical Zendo
« Reply #49 on: December 04, 2013, 08:49:03 pm »

Further observation: Whatever the extra term on top of that 50 period cycle, can also be simplified to a 25 period cycle that reverses at the end. The extra term from the 9th to the 34th is the same as the one from the 34th to 59th but in the opposite direction (so t(33)=t(35), t(32)=t(36) etc.)
Compare this to the property of nx^2, where n is any integer number.
« Last Edit: December 04, 2013, 08:50:08 pm by ghostofmars »
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