Circle Numbers

[GB]

Hi,

I am trying to solve a problem where I have a series of circles with numbers in them. Each circle, as it expands has 8 more numbers than the previous circle.


What am I trying solve? Lets say I select number eg 23 and I draw a line out from zero (the centre) through 23 and beyond. What I want to find is the closest number to 834 that will touch this line.


I am struggling with writing the right formula, but have some preliminary calculations. Note, the only numbers that come into perfect/obvious alignment are the lowest numbers in each circle (0,1,9,25...). See attached.


https://dl.dropbox.com/u/60464004/CircleNumbers.xlsx


Thanks in advance

regards

GB

 

[Faseeh]

Hi GB,


Its interesting problem, Let me try it!! :D


Regards,

 

[keymaster]

It would be on 14 circle, the closest number is 827.


See this file:


http://img.chandoo.org/playground/CircleNumbers.xlsx


How it works:


What we have here is, the circle sizes are an arithmetic series with 8 as gap.

so first circle has 8, then 16, then 24.

That means, for nth circle, the start would be 8*n*(n-1)/2 + 1 = 4*n*(n-1) + 1

Since the line crosses first circle on point 8, it has an angle of 315 (8-1)/8 * 360

Hence, the point on nth circle would be, 315/360 * circumference of circle + circle start


= 315/360 * (2*pi*r) + 4*n(n-1) + 1

where r is the radius of circle. Now we don't know the radius, but we do know how many dots are there on each circle. It is 8n

thus 8n = 2*pi*r


replacing 2*pi*r with this in above equation, you get:


=315/360 * 8n + 4n*(n-1) + 1

= 7n + 4n*(n-1) +1

So enter this in a cell (where n is 1,2,3,4....) and drag it down and you have the values for all circles.

 

[GB]

Hi Chandoo,

very impressive, thanks for the explanation. I'm sure I will learn a lot from this.


regards

GB