Drawing Circles
Consider the above diagram of two intersecting circles with two intersection points marked by red dots. You are to start on either one of the dots and draw the two circles without lifting the pen and without tracing the same line more than once.
How many different possibilities are there that you can draw these circles?
Clarification: Possibilities include starting on a different dot.
The answer is 48.
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1 solution
Consider 2 red dots as vertices v 1 , v 2 and it is connected by 4 edges.So now we have to find Eulerian paths from the diagram.
One path will be like this v 1 → v 2 → v 1 → v 2 → v 1 .For the first time there are 4 paths available then after chosing one path there are 3 paths available and in this way at last two and one path will be available.So number of ways 4 × 3 × 2 × 1 = 2 4
Here we have started with v 1 now if we start with v 2 then same number of paths will be accounted.
So, total number of paths will be 4 8