From the formula used to figure figure-skating scores to the calculus used to figure instantaneous velocities in a speed-skating race, arithmetic and math are part of every Winter Olympic event and every move Olympic athletes make on snow or ice. NSF-funded mathematician Edward Burger from Williams College explains some of the math you can see in Olympic sports, with assistance from figure-skating expert and sports scientist Deborah King of Ithaca College, and U.S. hockey player Ryan Miller.

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LESTER HOLT, Anchor: It might not be as obvious an Olympic sport as physics or materials engineering, but math, from simple arithmetic to calculus is part of every jump, every spin, every move the athletes make on snow or ice. Edward Burger, a professor in the Williams College Math Department, funded by the National Science Foundation explains why math counts.

The Winter Olympics: 2500 athletes competing in 86 winter sports events to win 252 medals – and those are only the ‘base’ numbers in the Games.

DR. EDWARD BURGER, Williams College: We see the numbers when we’re looking at scores and measurements, but math is all around us whenever there’s motion, whenever there’s quantities.

HOLT: …Like number of hockey players on the ice… number of times the puck goes into the net.

BURGER: Combine quantities, and that’s where addition comes into play.

HOLT: Addition is only part of the arithmetic used to score figure skating. Each element is assigned points, and judged on how well it’s performed.

DR. DEBORAH KING, Ithaca College: Triple axle might be worth three points; a spin might be worth two points.

HOLT: Judges also rate the overall quality and artistry of the skater’s program. There are nine judges – but the scores from only five will count.

KING: They throw out two of the judges’ scores just by random. And then we throw out the high and low. And once they do that, then they average the remaining scores together.

HOLT: To find an average, you add up a list of items – in this case, scores – then divide by the number of items – in this case 5 – to get the average, or mean. When all the averages are added up, the skater with the highest total wins the event.

Any Winter Olympic sport that is timed – downhill slaloms…bobsled runs…speed skating – is math in motion.

BURGER: People are moving at a particular rate, and that’s all mathematics. ‘Rate’ is the math slang for speed. Rate equals distance divided by time.

HOLT: Divide the distance of a race – 1,000-meters, for example – by the time it takes a short track speed-skater, like five-time Olympic medalist Apolo Ohno, to skate that distance – say, one minute, 24 seconds, or a total of 84 seconds… and you get the rate: 11.9 meters per second – that’s 42.8 kilometers per hour. Or – for those not on the metric system – 26.7 miles per hour.

All the skaters started the race at zero kilometers (or miles) per hour …How did Ohno win? Did he just skate faster than everybody else from the beginning?

BURGER: Asking who’s going the fastest, at the most fundamental level, is really the question of calculus. Because calculus tries to understand not just velocities, but instantaneous velocities – velocities at any one instant in time.

HOLT: One way to see that is as a graph – here, of two speed skaters in a 5000-meter race.

BURGER: So here’s the mathematical view of it. Consider time – which I’ll think of as the horizontal axis – and then we can think of distance away from the starting position in the vertical. So we have distance and time. Someone starts off a little bit slower and then accelerates – how would that look? It’d be a little bit less steep…and then increase. Or you can imagine someone sprinting out of the starting block and then tiring out. And that would look something like this: really, really steep at the beginning, but then slowing down. The steeper the curve, the faster the speed.

HOLT: Calculus shows which skater is going how fast at what particular point in time in the race.

BURGER: Here we can look at the entire race, from beginning to the end – and we can make judgment calls. For example, is it better to sprint at the beginning? Or is it better to actually start off a little slow, let everyone else get really, really cocky – the competition – and at the end, drive it up and win?

HOLT: And, to be the first to cross the finish line.

Lines are everywhere in the Olympics – finish lines…sight lines…line segments and angles – especially in hockey.

BURGER: It’s all geometry. It’s all lines. You’re measuring angles, and angles of incidence and angles of reflection.

HOLT: On defense, hockey players try to move attackers off to the side, reducing the angle of access to the net.

RYAN MILLER, U.S. Hockey: That angle just gets lower and lower and lower. He has less and less and less net, so his chances of scoring should be going down. And so it’s something that, in hockey, we are definitely using elementary geometry.

HOLT: Arithmetic…Algebra…Geometry…Calculus…

BURGER: Math is all around us. Did you catch that? I hope you did.