# Today Puzzle #613

## Puzzle No. 613â€“ Monday 18 November

A hockey stick is about one metre long, and it happens that this is about the same as the width of the lanes on an athletics track. If I toss the stick onto the track one hundred times how many times do I expect it to land crossing one of the lines?

Today’s #PuzzleForToday has been set by Hugh Hunt, Reader in Engineering Dynamics and Vibration at Trinity College, Cambridge

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In fact it's 200/π, so this is a neat experimental way to estimate a value for π.

Suppose the stick is at an angle Ñ² to the track (Ñ²=0 is parallel to the track). You can slide the stick a distance sinÑ² while touching one of the lines and then 1-sinÑ² is the distance you can move the stick in the space in between the lines. The probability of touching a line for a given Ñ² is therefore sinÑ². Now what about for all possible Ñ² between 0 and 180°? Well, we add up all the probabilities and take the average. Easiest to do that by integration. ∫ sinÑ² dÑ² = -cosÑ² evaluated between 0 and π which is 1-(-1)=2. But this is for Ñ² over the range of π so for the average we divide by π which gives the answer of 2/π.

Now I do 100 tosses so the expected number of line crossings is 200/π.

Why not try this out experimentally on the sports field? Just cut a bamboo stick to the same width as the lanes on your track and get tossing!