Two needles have the same length. One is straight; the other may be bent at will. Can a clever shape outwit chance?
The Bent Needle
Riddle statement
A floor is marked by parallel lines 10 centimetres apart.
You have two needles, each 10 centimetres long:
- the first remains straight;
- the second may be bent at as many points as you like, so that it consists of several straight segments. Once its shape has been chosen, it remains rigid.
Each needle is dropped at random onto the floor, with no orientation or position relative to the lines being favoured.
Count one intersection whenever a segment of the needle crosses a floor line. If several segments cross the same line, count every intersection separately.
Can you bend the second needle so that, on average, it produces fewer intersections than the straight one?
Additional challenge: calculate the average number of intersections per drop.