## Thursday, March 4, 2010

### Brian Pothier on Ulam's Conjecture

Today’s question comes from this week’s special guest, Brian Pothier.  In addition to being an NHL defenseman for the Washington Capitals, Brian may also be an amateur physicist.  Perhaps without realizing it, Brian suggested a question that contains some fairly deep physics:

“How many pucks can you fit in a hockey net?”

The packing of three-dimensional geometric objects has been an on-going topic of research from antiquity to the present day1.  Since we’re only concerned with an order of magnitude estimate, we won’t be treating this problem with the rigor that a mathematician might, but the odd shape of the goal makes finding a rigorous solution an interesting—or at the very least challenging—problem.

An official NHL hockey goal is 4 ft tall and 6 ft wide.  As mentioned above, the back of the net has a funny shape, but it looks about 2 ft deep.  This gives a total volume of ~1.4 m3 (42 ft3).

According to Wikipedia, a standard hockey puck is 1 in thick with a 3 in diameter giving a total volume of 116 cm3 (~7.06 in3).  If we stack them on top of each other in a hexagonal pattern, they’ll consume about 90.7% of the volume2.  From this and the volume above, we can compute the number of pucks that can fit in a net,

# of pucks = (packing fraction) · (vol. per net) · (vol. per puck)
= (0.907) · (1.4 m3) · (116 cm3)
= 9,300 pucks per net

That’s a lot of pucks, roughly equal to the number of goals that Brian’s team the Capitals have scored in their entire history.  Alex Ovechkin would have to keep up his current scoring pace for 150 years to score that many goals. Thanks for the question, Brian!