## Sunday, March 15, 2009

### Two kinds of relays: cross-country running and problem-solving

The photo at left shows Albany Area Math Circle alumna Alison Miller running in the third leg of yesterday's Turing Trail Relay, a 35.5 mile cross country race for 6 person teams from Ely to Cambridge (England) and back. The race is named after the mathematical logician Alan Turing, who regularly enjoyed running from Cambridge to Ely and back again along those same footpaths by the River Cam. Like many present-day mathematicians, he found distance running a source of inspiration, relaxation, and a good break from more deskbound pursuits. He was actually quite a serious distance runner decades before it became popular during the running boom of the 1970s. Turing came in fifth place in the British Olympic marathon trials in 1948. He was also a member of a running club that later helped Roger Bannister to break the four-minute mile.

Alison was running in the Turing Trail Relay as part of one of two teams from the Cambridge University Centre for Mathematical Sciences (CMS), which had the rather self-explanatory names of CMS1 and CMS2. Other teams had more intriguing names, (somewhat remniscent of names of some of the HMMT teams): the Muddy Mucky Munkeys, the Morley Marauders, the Sweatshop Social Running Group, the Abbey Ancients, Fetcheveryone 1, and the Thunderpants. The Cambridge Triathlon Alpha Males came in first place with a time of 3:29. CMS1 was a seriously competitive team and came in seventh place in 4:03. Alison was on the CMS2 team, whose goal was simply to finish the 35.5 mile relay, after having fun preparing together for it. which they did indeed manage to do--in a total time of 5:07, which placed them in 39th place out of 47 teams.

Albany Area Math Circle students also enjoy a very different kind of relay: the math relays at NYSML and ARML. The official relay competitions at those math meets are done in three person teams, who are sedately sitting at desks. There's no physical activity involved in the NYSML and ARML relays other than writing and passing slips of papers to the teammate behind you.  Each of the three students on a NYSML or ARML relay team receives a different problem. The answer to the first person's problem is an input to the second person's problem and the answer to the second person's problem, in turn, is an input to the third person's problem. The first student solves his or her problem and passes the answer back to the second student who uses that answer as an input into the problem s/he is working on, then passes the answer to the second problem on to the third student as an input into the final problem.

ARML also has a very fun (unofficial and just for fun!) Super Relay done by the whole 15 person team. In that case, the person in position #8 solves a problem using two inputs passed on by students working in two separate relay chains going forwards from position #1 to #8 and positions #15 backward to #8. You can try the problems in last year's ARML Super Relay here.
Some of the problems are more accessible than others--usually the teams are arranged with the most experienced students in the later positions.

Important things to keep in mind for success in math relays:

#1) The mysterious expression "TNYWR" often appears in relay problems. It's an abbreviation referring to "The Number You Will Receive."

#2) Unlike in a track or cross country relay race, you don't have to wait until you get handed the number by the person in front of you in order to start working on your problem. In fact, you shouldn't wait for the number. The correct strategy is to work on your problem even before you get your input, by simplifying it as much as possible in advance. Then when you get your "TNYWR" passed to you, you can quickly plug it into a simplified expression in order to generate the answer.

#3) You are allowed to pass back answers as often as you like. So you can pass back an answer, keep working, pass back another answer, and so on. You can even keep passing back the same answer, just to reassure the person behind you that you are really confident about your answer.

#4) The only thing you are allowed to pass back is the answer to your problem--no additional information. You are allowed to underline your answer in order to make the orientation clear (because otherwise, the person behind you might not be able to tell a 6 from 9.) But be careful about underlining--otherwise you might run the risk that what you intended to be a zero looks like a 10 when underlined and viewed sideways.

#5) Soemtimes the person in front of gives you a number that yields a clearly impossible input to your problem. For example, suppose your problem statement reads "Let ABC be a triangle whose hypotenuse has length equal to TNYWR and ....," but you the number you receive from the person in front of you is negative. In such a case, you are free to disregard TNYWR and take your best stab at guessing a plausible input to your problem. (It might even work! Sometimes the structure of your problem suggests a likely TNYWR.)