Around a circle i place 64 equally spaced points, so that there are 64x63 divided by2=2016 possible chords between these points. I draw some of these chords, but each chord cannot cut across more than one other chord. What is the maximum number of chords I can draw?
Could someone please give me a head start on how i should approach this problem. Its been bugging me for weeks, thanks.
To solve this problem, let's start by considering the number of chords that can be drawn from a single point.
If you have 64 equally spaced points around the circle, let's label them as P1, P2, P3, ..., P64. From a single point Pi, you can draw chords connecting it to all the other points except for the directly opposite point. So, for each point Pi, you have 63 possible chords to draw.
Now, let's consider the number of pairs of points that can be connected by a chord. Since each chord connects two points, we can calculate the total number of pairs using the formula n*(n-1)/2, where n is the total number of points.
In this case, we have 64 points, so the total number of chord pairs is 64 * (64-1) / 2 = 2016.
Since each chord connects two different points, we can draw a maximum of 2016 chords without any of them intersecting each other.
Keep in mind that when drawing the chords, you need to make sure that each chord does not cut across more than one other chord. This means that if two chords intersect, they can only intersect at one point.
I hope this helps you in approaching the problem!
To approach this problem, we need to understand the conditions and constraints given. We have a circle with 64 equally spaced points, and we want to draw chords between these points. However, each chord cannot cut across more than one other chord.
First, let's consider how many points there are in total on the circle. If we have 64 equally spaced points around the circle, this means that the entire circle is divided into 64 equal parts. Therefore, there are a total of 64 points.
Now, let's think about how many chords can be drawn between these points. Each chord requires two points, and we want to avoid having chords that cut across more than one other chord. One way to approach this is to start drawing chords and count how many can be drawn without violating the constraint. However, this could be time-consuming and may not guarantee the maximum number of chords.
Alternatively, we can use a combinatorial approach. Since each chord requires two points, we can select 2 points from the total of 64 points. Using the formula for combinations, denoted as nCr, where n is the total number of elements and r is the number of elements to be selected, we can calculate the total number of chord possibilities.
nCr = n! / (r! * (n-r)!)
In this case, n = 64 (the total number of points) and r = 2 (the number of points required for each chord).
64C2 = 64! / (2! * (64-2)!) = 64! / (2! * 62!)
Now, let's calculate this value using Python code:
```python
import math
n = 64
r = 2
chord_count = math.factorial(n) // (math.factorial(r) * math.factorial(n - r))
print(chord_count)
```
Running this code will give us the result:
2016
Therefore, the maximum number of chords that can be drawn between these 64 equally spaced points, without any chord crossing another chord, is 2016.
Remember to consider the symmetry of the circle when counting chords. That is, if you draw a chord from point A to point B, you don't need to draw another chord from point B to point A, as it would be redundant.
try nesting polygons
... going from the outside in
connecting adjacent points gives 64 sides (chords)
every other point gives 32 sides
every 3rd point. etc...