Happy Friday Math Gang;
I can't seem to wrap my head around this one...
If two circles have at most 2 places of intersections, 3 circles have at most 6 places of intersection, and so on... How many places of intersection do 100 circles have?
Well... I can't even draw them quite right.
I am not certain if they have to only intersect the way The Olympic Circles do, or if they can be embedded in one another....
I think 4 circles have 12 places of intersection,
5 circles have 14,
But they are only guesses and I can not derive a pattern rule...
Any thoughts would be greatly appreciated.
Yours in Mathematical fun, Ms Pi_3.141592653589793238462643383...
https://math.stackexchange.com/questions/1241472/the-greatest-number-of-points-of-intersection-of-n-circles-and-m-straight-lines Is a similar question, and most interesting graphics
Totally AWESOME graphics!
Thank you so much!
I hunted, and hunted and found nothing as lovely as that!
Thanks again,
Happy Friday Night!
Yours in Mathematical fun, Ms_Pi_3.1415926535897933238462643383...
A circle can cut another circle in at most 2 places.
So any new circle cuts
Think along the lines of the "handshake" problems
2 people can shake hands in 1 way , C(2,2)
2 circles can intersect in 2 ways ---->2x1 = 2
3 people can shake hands in 3 way , C(3,2)
3 circles can intersect in 6 ways ----> 2x3 = 6
4 people can shake hands in 6 ways, C(4,2)
4 circles can intersect in 12 ways ---> 2x6 = 12
5 people can shake hands in 10 ways, C(5,2)
5 circles can shake hands in 20 ways ----> 2x10 = 20 ..... hard to draw the circles and count
(how did you get 14 ?)
.....
Some nice diagrams showing 5 circles intersection
https://www.google.ca/search?rlz=1C5CHFA_enCA690CA690&biw=1651&bih=881&tbm=isch&sa=1&ei=IRtJW-7tH4iRjwSiwIT4DA&q=venn+diagram+5+circles&oq=venn+5+circles&gs_l=img.1.0.0i7i30k1l2.72863.73524.0.76458.4.4.0.0.0.0.228.488.3j0j1.4.0....0...1c.1.64.img..0.4.486....0.4uikel5lQyQ#imgrc=_
So for n circles, my projection is 2xC(n,2)
Thank you so much Math Gang for your outstanding tutorials on my problem! Yippeeee! I do love a good math problem.
Yours in Mathematical fun, Mrs Pi 3.141592653589793238462643383...
https://www.google.com/search?client=firefox-b-1&biw=1384&bih=802&tbm=isch&sa=1&ei=VqVoW8-hC8vAsAW64JHIDA&q=inscribed+isosceles+triangle&oq=inscribed+isosceles+triangle&gs_l=img.3..0i30k1j0i8i30k1l2j0i24k1.291489.294036.0.294233.9.9.0.0.0.0.121.771.8j1.9.0....0...1c.1.64.img..1.8.698...0i7i30k1j0i7i5i30k1j0i8i7i30k1.0.INqJV4wWOHI#imgrc=55TcSC0EBwUQ3M:
Happy Friday Ms. Pi_3.141592653589793238462643383...!
The problem you are trying to solve involves finding the number of intersections between a certain number of circles. To understand and solve this problem, let's break it down step by step:
1. Start by analyzing the number of intersections between two circles. You mentioned that two circles have at most 2 places of intersection. This is correct. Two circles can either not intersect at all, intersect at a single point (tangent) or intersect at two points.
2. Now, let's move on to three circles. You mentioned that three circles have at most 6 places of intersection. To find the number of intersections, we first need to consider pairs of circles.
- Pair 1: Circle 1 and Circle 2 have at most 2 places of intersection.
- Pair 2: Circle 1 and Circle 3 have at most 2 places of intersection.
- Pair 3: Circle 2 and Circle 3 have at most 2 places of intersection.
In total, we have 6 possible places of intersection.
3. To find the number of intersections for four circles, let's continue the same approach.
- Pair 1: Circle 1 and Circle 2 have at most 2 places of intersection.
- Pair 2: Circle 1 and Circle 3 have at most 2 places of intersection.
- Pair 3: Circle 1 and Circle 4 have at most 2 places of intersection.
- Pair 4: Circle 2 and Circle 3 have at most 2 places of intersection.
- Pair 5: Circle 2 and Circle 4 have at most 2 places of intersection.
- Pair 6: Circle 3 and Circle 4 have at most 2 places of intersection.
In total, we have 12 possible places of intersection.
4. Continuing this pattern, we can observe that the number of intersections for n circles can be found using the formula:
Number of intersections = (n-1) + (n-2) + (n-3) + ... + 2 + 1
We can simplify this formula using the summation notation:
Number of intersections = Σ(i=1 to n-1) i
This summation formula can be evaluated as:
Number of intersections = (n-1)(n-1+1)/2 = (n-1)n/2
Using this formula, we can find the number of intersections for 100 circles:
Number of intersections = (100-1) * 100 / 2 = 99 * 100 / 2 = 4950
Therefore, 100 circles can have at most 4950 places of intersection.
I hope this explanation helps you understand the problem better. Let me know if you have any further questions!