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        riddles >> what happened >> A royal progression
        
(Message started by: towr on Dec 21st, 2016, 11:44am)
Title: A royal progression
Post by towr on Dec 21st, 2016, 11:44am
A king offered a (supposedly) wise man one wish. (http://www.smbc-comics.com/comic/the-other-side-of-the-chessboard-2)

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!" ([edit]8,256[edit])

The apparently not so wise man left confused and disappointed, wondering what had happened.
Title: Re: A royal progression
Post by Grimbal on Dec 23rd, 2016, 5:19am
I don't know the answer, but I realized the sum of 0 to 63 is 2016.
Pfew... just in time.
Title: Re: A royal progression
Post by towr on Dec 23rd, 2016, 6:05am

on 12/23/16 at 05:19:39, 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.)
Title: Re: A royal progression
Post by rloginunix on Dec 23rd, 2016, 8:35am
My first guestimate was:

an = 2n mod x

No cigar for x http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gif [65, 8257]

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


[e]
n must be x
[/e]
Title: Re: A royal progression
Post by towr on Dec 23rd, 2016, 1:35pm
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).
Title: Re: A royal progression
Post by rloginunix on Dec 23rd, 2016, 1:45pm
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".
Title: Re: A royal progression
Post by towr on Dec 24th, 2016, 12:14am
Well, that's one of the six parts, so it's a good start (or end).
Title: Re: A royal progression
Post by towr on Dec 29th, 2016, 10:57pm
Clues for the first two sequences: pizza and cake
Title: Re: A royal progression
Post by rloginunix on Dec 30th, 2016, 3:19pm
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.
Title: Re: A royal progression
Post by towr on Dec 31st, 2016, 12:12am
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)
Title: Re: A royal progression
Post by rloginunix on Dec 31st, 2016, 7:01am
Pfew! One down, five to go.

Number 2 of 6 seems to be [hide]A000125[/hide], click (http://oeis.org/A000125).
Title: Re: A royal progression
Post by towr on Jan 1st, 2017, 5:46am
Yup, that's correct.
The first four are all related, so the other three should be easy to find.
Title: Re: A royal progression
Post by rloginunix on Jan 1st, 2017, 10:43am
For number 3 of 6 - [hide]A000127[/hide] here (https://oeis.org/A000127), summing to [hide]8,303,632[/hide] grains of rice?
Title: Re: A royal progression
Post by towr on Jan 1st, 2017, 11:05am
Yup.
Title: Re: A royal progression
Post by rloginunix on Jan 7th, 2017, 6:57pm
For number 4 of 6 - [hide]A006261[/hide] here (http://oeis.org/A006261), summing to [hide]83,278,000   (91,071,217)[/hide]?
Title: Re: A royal progression
Post by towr on Jan 8th, 2017, 9:21am
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.
Title: Re: A royal progression
Post by rloginunix on Jan 8th, 2017, 9:44am
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 [hide]83,278,000[/hide].


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