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towr
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 A royal progression   « on: Dec 21st, 2016, 11:44am » Quote Modify

A king offered a (supposedly) wise man one wish.

The wise man asked, "I want a single grain on the first square of the chessboard. Two on the second square, four on the third and so on"

The king considered his request for a while, picked up pen and paper and after a few scribbles asked the wise man, are you sure you're satisfied with 8,213 grains of rice?

The wise men blinked. "Err, sire, I expect there to be 8 grains on the 4th square"

"Ah, so 679,120 grains. Roughly 10 and a half kilo's of rice, then?"

"No, no, the fifth square has 16 grains of rice on it"

The king looked puzzled for a moment and scribbled some more on his piece of paper. "So, 130 kilos of rice?"

"No, Sire. There should be 32 grains of rice on the 6th square"

"This would be easier if you just said how much rice you wanted from the start, instead of making me guess. How about 1300 kilos?"

"Please, sire, don't you see the obvious progression, 1,2,4,8,16,32, the next must be 64!"

The king sits back in his chair and thinks for a moment. After a few more calculations he looks the wise man in the eyes, a slight smirk on his face. "Ah, I see, I see. I was on the wrong track. You modestly asked only for enough rice for the next two weeks. 308,346 grains, to be exact. Right?"

The wise man was taken aback, the amount of rice on the board had taken a sudden turn for the worse, it had been increasing by leaps up to now, and suddenly only a fraction was left. "I, err.." The wise man hesitated. "Umm.. the.. the top row, they're all increasing powers of two my lord. Don't you see?"

The king sighed. "Fine, then." And with a solemn finality declared "3,415 grains of rice it is. Now leave!" (8,256)

The apparently not so wise man left confused and disappointed, wondering what had happened.
 « Last Edit: Jan 8th, 2017, 9:22am by towr » IP Logged

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Grimbal
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 Re: A royal progression   « Reply #1 on: Dec 23rd, 2016, 5:19am » Quote Modify

I don't know the answer, but I realized the sum of 0 to 63 is 2016.
Pfew... just in time.
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towr
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 Re: A royal progression   « Reply #2 on: Dec 23rd, 2016, 6:05am » Quote Modify

on Dec 23rd, 2016, 5:19am, Grimbal wrote:
 I don't know the answer, but I realized the sum of 0 to 63 is 2016.
And if you add its square and its cube, you have each digit from 0 to 9 once! And surprisingly, the sorted digits divide the number itself (2016 = (0)126 * 16).

As for the puzzle, in case it's not clear, it's about finding what sequences the king had in mind (as opposed to the obvious one the wiseguy was thinking of).
And you can find them all with the last link in my signature. Just need to find the ones that add up.

(There might be better / more interesting ones; e.g. the first ones are all similar. And it might have been nice to plateau the amount of rice for a few steps before going down.)
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 Re: A royal progression   « Reply #3 on: Dec 23rd, 2016, 8:35am » Quote Modify

My first guestimate was:

an = 2n mod x

No cigar for x [65, 8257]

(the closest it came was for n x = 303 summing to 8,259)

[e]
n must be x
[/e]
 « Last Edit: Dec 23rd, 2016, 1:41pm by rloginunix » IP Logged
towr
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 Re: A royal progression   « Reply #4 on: Dec 23rd, 2016, 1:35pm » Quote Modify

Hmm, well at each step the king's sequence deviates at the next "unrevealed" square from what the wise man had in mind. Otherwise the king wouldn't have to try another sequence when the wise man reveals the next square.

So with the scheme you propose, for the first guess, x would have to be in [5,8] to avoid getting 8 in the 4th square.

Anyway, all sequences are strictly increasing (in one way or another ), except maybe the fifth because I can't quite remember it at the moment (but I have it written down somewhere).
 « Last Edit: Dec 23rd, 2016, 1:35pm by towr » IP Logged

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 Re: A royal progression   « Reply #5 on: Dec 23rd, 2016, 1:45pm » Quote Modify

I see. When I read this I interpreted it as "given first eight terms, which happen to be the first exact powers of two, find the sequence whose 64 terms sum to 8256".
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towr
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 Re: A royal progression   « Reply #6 on: Dec 24th, 2016, 12:14am » Quote Modify

Well, that's one of the six parts, so it's a good start (or end).
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towr
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 Re: A royal progression   « Reply #7 on: Dec 29th, 2016, 10:57pm » Quote Modify

Clues for the first two sequences: pizza and cake
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 Re: A royal progression   « Reply #8 on: Dec 30th, 2016, 3:19pm » Quote Modify

Sorry, don't mean to sound nagging but every time I click the "Integer sequence DB" in the signature I get "503 Service Unavailable". Is this another puzzle for us to solve or something is off with the link? Is OEIS a reasonable alternative?

Thanks.
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 Re: A royal progression   « Reply #9 on: Dec 31st, 2016, 12:12am » Quote Modify

Oops, I guess that link has died long ago. It is in fact where (what  is now) OEIS used to reside*.
I've updated my signature, thanks for pointing it out.

(And it used to redirect, see e.g. https://web.archive.org/web/20111102001711/http://www.research.att.com/~ njas/sequences/index.html in case you're wondering how to solve the riddle of dead links)
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 Re: A royal progression   « Reply #10 on: Dec 31st, 2016, 7:01am » Quote Modify

Pfew! One down, five to go.

Number 2 of 6 seems to be A000125, click.
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 Re: A royal progression   « Reply #11 on: Jan 1st, 2017, 5:46am » Quote Modify

Yup, that's correct.
The first four are all related, so the other three should be easy to find.
 « Last Edit: Jan 1st, 2017, 5:47am by towr » IP Logged

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 Re: A royal progression   « Reply #12 on: Jan 1st, 2017, 10:43am » Quote Modify

For number 3 of 6 - A000127 here, summing to 8,303,632 grains of rice?
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 Re: A royal progression   « Reply #13 on: Jan 1st, 2017, 11:05am » Quote Modify

Yup.
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 Re: A royal progression   « Reply #14 on: Jan 7th, 2017, 6:57pm » Quote Modify

For number 4 of 6 - A006261 here, summing to 83,278,000   (91,071,217)?
 « Last Edit: Jan 8th, 2017, 9:45am by rloginunix » IP Logged
towr
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 Re: A royal progression   « Reply #15 on: Jan 8th, 2017, 9:21am » Quote Modify

It's the right sequence, but I get a different sum. (That's not too important, though)

The 5th and 6th are something completely different. (Because otherwise it would have kept going up.)

... huh, and I screwed up the 6th. It should be 3415, I'll go fix that.
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 Re: A royal progression   « Reply #16 on: Jan 8th, 2017, 9:44am » Quote Modify

Yup - pilot error: the sequence' numbering starts with a "0", not "1", so the last term goes under "63" - not "64". I'll fix that.

The correct sum for number 4 is 83,278,000.
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