How many ways are there to trace this figure without lifting your pencil or adding anything?
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You have to start from the straight line from either vertex as they both have odd connections(YOU C.ANT START ON THE CIRCLE , you cant trace it then)if we start from the left one we will still end up on the right one so , the ways from right vertex = ways from left vertex.
So now we are on right vertex & we have 2 choices , either to go up or go down on the circle , lets say we go up & then on next intersection we have same situation. So we have two choices on each intersection & we have 4 of these so total ways = 2x2x2x2 = 16 ( When coming back we take the path we left so there is no choice when coming back) . So 16 from right vertex & 16 from left one so the answer is 32.