Problem 354

[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

[jaap]

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

[rockly]

Does L has to be an integer or it can be a irrational number as well?

[sivakd]

The problem says "For a positive real number L"

[rockly]

Thanks

[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]

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

[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]

yes

[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.

[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. :/

[jaap]

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).

[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.

[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]

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]

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.

[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]

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.

[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.

[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.