Happy Ending Problem

HAPPY ENDING PROBLEM !!!

ENJOY THE BEAUTY & MYSTERY OF RANDOMNESS, PROBABILITY AND GEOMETRY !!!

Five green dots are placed at random on the screen.

The dots are generated randomly on different regions of the screen.

Suppose that all the 5 dots are not in a line and the dots are separated from each other so that you can distinguish the dots and click on these.

You should always be able to connect four of them to create a convex quadrilateral, which is a shape with four sides where all of the corners are less than 180 degrees.

As per Wikipedia :

"A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set.
In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees."

The moral of the theorem is that you'll always be able to create a convex quadrilateral with five random dots, regardless of where those dots are positioned.

The moral of the story is that how it works for four sides.

But for a pentagon, 9 dots are required.For a hexagon, 17 dots are required.

But beyond that, we still don't know.

It's a mystery how many dots are required to create a heptagon or any larger shapes.

There might be a formula to tell us how many dots are required for any shape.
Mathematicians suspect the equation is M =1 + 2^(N - 2), where M is the number of dots and N is the number of sides in the shape. Here ^ denotes power.

This simple game deals with only 5 dots i.e. the case for a convex quadrilateral.

This game may run slow on some devices.

BUG :

*** In the instructions formula is wrongly written as M=1+2N-2 instead of M=1+2^(N-2).

This game is ABSOLUTELY FREE, has NO-ADS or NO IN-APP PURCHASES.

*** In case of any bug or any misinformation, please email me.