Problem 354

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BostonBear
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Problem 354

Post by BostonBear »

This post is in regards to calculating B(1111111111). I had no problems finding out the method to calculate distances between the centers of 2 hexagons and creating a script to calculate B(sqrt(3)) and B(sqrt(21)) was straight forwards. But finding a distance between 2 hexagons that is equal to 111111111 has proven to be problematic. Its clear that when you break it into x**2 and y**2 = 111111111 that x has to be either an integer or a mutiple .5. Yet brute forcing it I was unable to find any solution whatsoever that lets 111111111 be a valid solution. I took into account rounding errors of square roots, etc but still no dice. If somebody could give me a hint on what I am doing wrong or give me one coordinate set that leads to that distance, I would be greatly appreciative.

Mike :D
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jaap
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Re: Problem 354

Post by jaap »

Can you find two hexagon centres that are exactly distance 3 apart?
Think about how would that might help.
rockly
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Re: Problem 354

Post by rockly »

Does L has to be an integer or it can be a irrational number as well?
sivakd
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Re: Problem 354

Post by sivakd »

The problem says "For a positive real number L"
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rockly
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Re: Problem 354

Post by rockly »

Thanks
JLes
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Re: Problem 354

Post by JLes »

I have found a general formula for L (B (L) = 450). It seems to be more than one solution below 5*10^11. What number must be entered? Minimal? Maximum?
mynameisalreadytaken
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Re: Problem 354

Post by mynameisalreadytaken »

As the Problem states: You have to find the number of L <= 5 * 10^11, which hold B(L) = 450.
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spacetweek
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Re: Problem 354

Post by spacetweek »

It says to calculate for L <= 5.10^11
I presume in this context that dot means multiply?
So the value for L is 500 000 000 000?
sivakd
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Re: Problem 354

Post by sivakd »

yes
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puzzle is a euphemism for lack of clarity
Duality
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Re: Problem 354

Post by Duality »

I can find which ones are exactly 3 away, but it seems like the number of circles I can toss out within a hex-ring to capture centers increases as I go from one hex-ring to the next.
Last edited by Duality on Mon Nov 28, 2011 4:00 am, edited 1 time in total.
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mctrafik
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Re: Problem 354

Post by mctrafik »

I don't get how B(sqrt(21)) = 12
I make a grid. Then draw a circle going through the 7th hexagon directly above the queen and it doesn't intersect anything but the other 5 symmetrical bees. :/
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jaap
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Re: Problem 354

Post by jaap »

mctrafik wrote:I don't get how B(sqrt(21)) = 12
I make a grid. Then draw a circle going through the 7th hexagon directly above the queen and it doesn't intersect anything but the other 5 symmetrical bees. :/
Have you looked carefully at the picture in the problem? It shows a length of sqrt(21).
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Lord_Farin
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Re: Problem 354

Post by Lord_Farin »

It was helpful for me to realise that the side lengths of the hexagon are 1. This is not the same as the distance between two adjacent hexagons' center points being 2.
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Aristidis
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Re: Problem 354

Post by Aristidis »

Can somebody please clarify this to me:

-A) Must we enter the total number (i) of Ls which satisfy B( L(1-i) ) = 450 (Li : integer or real)? For example, if there were i=100 different Ls, integers or real numbers, and for each L is true that B(L(i)) = 450, then the answer would be 100.
OR
-B) we seek something else (if yes please describe what)

Thanks in advance
Aris
gnanasenthil654321
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Re: Problem 354

Post by gnanasenthil654321 »

In this problem , i can understand B(sqrt(3) = 6 and B(sqrt(12)), but how B(L) (any valid L) be any number other than 6 and 12 , because

For example let us take a hexagon whose sides are at a distance 'a' from the center, now let us chose a point at distance 'b' from the center (this point is not the mid point of any side nor any vertex of the hexagon) , so is it not right to say that there are only 12 such points on the hexagon???

Kindly explain

thanks in advance
thundre
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Re: Problem 354

Post by thundre »

gnanasenthil654321 wrote:In this problem , i can understand B(sqrt(3) = 6 and B(sqrt(12)), but how B(L) (any valid L) be any number other than 6 and 12 , because

For example let us take a hexagon whose sides are at a distance 'a' from the center, now let us chose a point at distance 'b' from the center (this point is not the mid point of any side nor any vertex of the hexagon) , so is it not right to say that there are only 12 such points on the hexagon???
I think you're being sidetracked by symmetry.

You can tile the plane with concentric rings of hexagonal cells. Each ring has 6-way symmetry, and any hexagon you choose will have 5 or 11 symmetric brothers in that ring which are the same distance away from the center. However, there can be other symmetric groups of 6 or 12 which are the same distance from the center but part of other rings.
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AndyNovo
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Re: Problem 354

Post by AndyNovo »

Hi all,
First time post here, I feel like I've solved the problem mathematically, and I'm not getting the right number. So I'm looking to see if I'm missing something or not without giving away hints or violating ToS. First here are a few values of L which I believe give B(L) == 450, 44760094341, 139178767*sqrt(3), 179936733613*sqrt(3)

I have between 20 and 30 million such numbers of the right size.

I have the number of solutions for L <= 5*10^9 at 309767. Is this close? Does someone who has the right answer see something mathematical I'm missing or is this a painful bug?
thundre
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Re: Problem 354

Post by thundre »

AndyNovo wrote:First here are a few values of L which I believe give B(L) == 450, 44760094341, 139178767*sqrt(3), 179936733613*sqrt(3)
Those examples look good, but your counts are too low. You must be missing some of the Ls.
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Djinx
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Re: Problem 354

Post by Djinx »

For a L<=5*10^9, I get the answer as 1133713 and for L<=5*10^10 as 11623092. Could someone comment on it?

EDIT: Never mind. Those answers are wrong. Fixed a bug and finally got it.
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dg_no_9
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Re: Problem 354

Post by dg_no_9 »

In this problem , i can understand B(sqrt(3) = 6 and B(sqrt(12)), but how B(L) (any valid L) be any number other than 6 and 12 , because

For example let us take a hexagon whose sides are at a distance 'a' from the center, now let us chose a point at distance 'b' from the center (this point is not the mid point of any side nor any vertex of the hexagon) , so is it not right to say that there are only 12 such points on the hexagon???

Kindly explain
I'm also thinking the same thing as you think, please somebody explain it.
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