[HoustonHokie]

In January every year, Cub Scout Pack 123 holds its annual Pinewood Derby competition. The competition (for those who've never seen a Pinewood Derby) is held to determine which Cub Scout (and/or his dad ) can build the fastest car from a block of wood and some standard wheels. At Cub Scout Pack 123, the Derby is held on a track with 6 lanes. Cars race against one another in heats, and the competition is set up so that each car gets one run in each lane of the track. The car with the lowest average time across all lanes is declared the winner. Prizes are given to the top 3 cars in each Den and the top 3 overall.

For many years, the competition went on and was running relatively smoothly, except for the normal little hiccups that seem to happen at every Derby. But two years ago, the number of cars entered in the race ballooned. The Derby, which used to take 2 hours to run, dragged on for 4 hours that year. And if you've ever been in a room with a whole bunch of bored Cub Scouts who don't have anything to do except wait for their car to make a trip down the track -- well, let's just say they don't have the greatest attention span in the world, and general chaos ensued. By the time the exhausted Cub Master left the Derby grounds that afternoon, he was vowing that next year would be different.

And it was. The Derby Committee met, and they decided to adopt new rules for the next year. Instead of running all cars in all 6 lanes, each car would race just twice. The lane assignments would be random so that no one would claim bias or preferential treatment. And, as in years past, the winners would be the car with the lowest average times.

So, last year, the Derby was held with the new rules in place. And the competition, even though it had more cars than the previous year, took less time and seemed to go more smoothly. That is, until the winners were announced. It seems that Lane 4 had more winners than all the other lanes put together. One particularly irate parent complained that his son, whose car raced only in Lanes 2 & 6, was only 0.01 seconds behind the winners in Lane 4. They claimed that if their son's car had raced in Lane 4, it would have placed among the winners.

So the Derby Committee had a new problem: how to ensure a fair race with the minimum number of heats. As you are the aforementioned irate parent, the Committee has come to you for suggestions of new rules to solve their problem. Begrudgingly, you accept their offer.

You look into the race results from the past couple of years, and you make some findings as follows:

1. Although Lane 4 has the most winners, Lanes 1 & 2 also have a significant amount of fast times.
   
2. Each lane of the track produces the fastest time for some cars, but not for others.
   
3. Times across all lanes generally do not vary by more than 5% for invidual cars.
   
4. In years past, there was an average of 60 cars entered in the Derby from 6 Dens (10 Cub Souts in each Den).
   
5. In the last two years, there was an average of 120 cars entered in the Derby from 10 Dens (12 Cub Scouts in each Den).
   
6. It is unknown how many Dens or cars will be in the competition this year.
   
7. The Committee wants to keep the award structure in place as it has been in the past, with prizes to the top 3 cars in each Den and the top 3 overall.
   

What do you tell the Committee?

[Guest]

You look into the race results from the past couple of years, and you make some findings as follows:

1. Although Lane 4 has the most winners, Lanes 1 & 2 also have a significant amount of fast times.
   
2. Each lane of the track produces the fastest time for some cars, but not for others.
   
3. Times across all lanes generally do not vary by more than 5% for invidual cars.
   
4. In years past, there was an average of 60 cars entered in the Derby from 6 Dens (10 Cub Souts in each Den).
   
5. In the last two years, there was an average of 120 cars entered in the Derby from 10 Dens (12 Cub Scouts in each Den).
   
6. It is unknown how many Dens or cars will be in the competition this year.
   
7. The Committee wants to keep the award structure in place as it has been in the past, with prizes to the top 3 cars in each Den and the top 3 overall.
   

What do you tell the Committee?

"There's no defense against xxxx-pot luck."

Split the competitors into groups of odds and evens. Each car races each odd or even track-- three races per car. Throw away the highest and lowest times for each car.

Race the top three of the odds against the top three of the evens, this time switching tracks. As before, use only the median of each of the three races for each of the six cars. Determine the winners by the average of their two middle scores (one for odd, the other for even).

[bonanova]

Develop an algorithm for finding the top three cars in a group of ten, where the lane assignments are drawn out of a hat. Base the algorithm on eliminating any car when its time [on its randomly chosen lane] is more than 5% longer than any other three cars [on their randomly chosen lanes] in the group.

If the 5% is a red herring, then develop any algorithm for finding the three fastest of a group of ten. Repeat for each den.

Say this takes n races using the six lanes.

Repeat for each den.

After 10n races you have the top 3 for each den and top 3 for the troop.

If n is between 4-6 you're no worse off that the 60 races run initially or the 40 races run last year.

[HoustonHokie]

Split the competitors into groups of odds and evens. Each car races each odd or even track-- three races per car. Throw away the highest and lowest times for each car.

Race the top three of the odds against the top three of the evens, this time switching tracks. As before, use only the median of each of the three races for each of the six cars. Determine the winners by the average of their two middle scores (one for odd, the other for even).

"There's no defense against xxxx-pot luck."

Actually, that's what they told you to begin with... and then you had to go and be all irate and stuff

Suppose Cars A & B raced in the odd lanes and Car A beat Car B by 0.02 s every time. But, if they had raced in the even lanes, Car B would have beaten Car A by 0.03 s every time. If Car A was third on the odd lanes, Car B would lose the opportunity to show itself as the better car overall. Can this situation be prevented?

It would probably be better to use the mean rather than the median. Imagine the following race times for Cars A & B again:

Race 1: A - 2.600 B - 2.595

Race 2: A - 2.608 B - 2.610

Race 3: A - 2.594 B - 2.598

Not only did Car A beat Car B head-to-head 2 of the 3 times they raced, but Car A had the lowest mean time. But if you took the median, Car B would be placed ahead of Car A. The idea is to get the same race results as you wuold have with the old rules, but take fewer heats to do so...

I like your general ideas, but you probably have some work to do on the details. Keep going...

[HoustonHokie]

Develop an algorithm for finding the top three cars in a group of ten, where the lane assignments are drawn out of a hat. Base the algorithm on eliminating any car when its time [on its randomly chosen lane] is more than 5% longer than any other three cars [on their randomly chosen lanes] in the group.

If the 5% is a red herring, then develop any algorithm for finding the three fastest of a group of ten. Repeat for each den.

Say this takes n races using the six lanes.

Repeat for each den.

After 10n races you have the top 3 for each den and top 3 for the troop.

If n is between 4-6 you're no worse off that the 60 races run initially or the 40 races run last year.

The 5% is not entirely a red herring. I threw that in there to make sure no one was trying to deal with a car that looked like a winner in 5 of the 6 lanes, but performed terribly in the 6th lane - enough to move it off the leader board entirely. 5% seemed appropriate for the races I've seen.

You're on the path to the solution I was looking at. A couple questions to hone in on the answer:

1. Is there an optimal grouping size for your comparisons?

2. Can you define boundaries for n?

[Guest]

Build a standard and simple car, make it the default car and run with it 5 times in each line. Measure the times and make alterations in the lines to make them padronized. Then you could use just 1 run for each player.

[Guest]

Build a standard and simple car, make it the default car and run with it 5 times in each line. Measure the times and make alterations in the lines to make them padronized. Then you could use just 1 run for each player.

My kids have raced cars in a couple of similar derbies, and I've helped in hosting one. There really is a fair bit of randomness to contend with-- alignment of the car at the starting position, fish-tail effect as the tires ping-pong from one side to another, alignment of the wheels in their pins after each handling, loss of dry-lubrication with each race, whether a wheel nicks a particular imperfection on a lane, the technique used to release the cars at the top of the track, variances in clearance from track to track, etc. In short, your test car may have a nice tight grouping of times on one lane, and a wide scattering of times on another. It's a good idea, mind you, but on well-used (or poorly stored) equipment, the variances can be quite noteworthy.

[Guest]

Suppose Cars A & B raced in the odd lanes and Car A beat Car B by 0.02 s every time. But, if they had raced in the even lanes, Car B would have beaten Car A by 0.03 s every time. If Car A was third on the odd lanes, Car B would lose the opportunity to show itself as the better car overall. Can this situation be prevented?

It would probably be better to use the mean rather than the median. Imagine the following race times for Cars A & B again:

Race 1: A - 2.600 B - 2.595

Race 2: A - 2.608 B - 2.610

Race 3: A - 2.594 B - 2.598

Not only did Car A beat Car B head-to-head 2 of the 3 times they raced, but Car A had the lowest mean time. But if you took the median, Car B would be placed ahead of Car A. The idea is to get the same race results as you wuold have with the old rules, but take fewer heats to do so...

I like your general ideas, but you probably have some work to do on the details. Keep going...

I think you're right-- a mean time would be better. Plus, people like seeing records being broken, so throwing away the fastest time would be kind of a let-down.

I'm still rather fond of the odd-even split, and think it would be judged as fair by even the most competitive-minded parents. The question of whether car A would be consistently better than car B on odd tracks, and consistently worse on even tracks seems unlikely and even irrelevant with respect to fairness. Given the constraints of those three lanes, it would still be a fair race. To wonder whether a car could have raced more favorably than a faster car in your three lanes if the race was performed in the other three lanes is about as useful as wondering if you could have raced faster on an entirely different track. Three lanes or six lanes shouldn't matter-- you either outperformed your peers on the same set of lanes, or you didn't. The top three cars (plus three others) would still have raced once in all six lanes.

If your goal is to minimize the number of cars that race in all six lanes, you could also have a sort of a grim-reaper elimination criteria. If you see a car whose times are so low that even matching the fastest times couldn't qualify him for a ribbon, than you just pull his car from the race. I suppose that's where your 5% might come in... if you don't have the time for all cars to run in all lanes, then the cuts have to come from somewhere.

I'm not sure I'd get the math right, but you could determine an arbitrary number of contestants from each group to race in the opposite lanes and eliminate everyone else after their first three races. It might soften the blow to explain that once you're down by 10% below the pack, there's just no recovering, but ultimately, you're trying to save time, and the most predictable way is not by percentages, but by number of contestestants.

Edited January 27, 2009 by Phatfingers

[Guest]

Do the race as normal but take the 3 highest ranking per lane. For the next heat mix them up into different lanes from their original lane and compare their time to the one who raced in the lane before. That way you can see if a lane has a bias and adjust the averages accordingly

[Guest]

By coincidence, two of my boys just raced their cars in an Awana Pinewood Derby event. They, eh, got ribbons for participating... and had a good time with their friends. This year, having your car make it completely down the length of the track put you ahead of about a third of the competitors (they both averaged 2nd place per heat). Also of note was that a substantial amount of sawdust was swept from the track between age groups left by cars with freshly-drilled holes to prevent disqualification from being overweight. The first cars down had a slight advantage. It wouldn't have made much difference for my kids times, but the difference between first and second place can be a pretty slim margin.

The people running it were pretty experienced-- it was their fifth annual race at that church. Their main strategy was to rely heavily on a computer to record the times and tabulate the rankings, and to have a volunteer for every two cars (one in each hand) getting ready for the next heat. By the time the cars were picked up at the bottom of the track, the next wave was ready to roll. As it happens, there were only four lanes and about 50 cars. I thought about the impact of reducing the number of lanes, but that doesn't help any-- it's the same number of heats either way.

[Guest]

Round one should be randomly selected lanes for each car. Cars should then be seated after Round 1 so that the top two in each lane all use the fastest lane in Round 2. The 3-4 finishers in each lane will then race in the second fastest lane. This pattern will continue and the lowest times are then given their awards after the two rounds.

[Guest]

Round one should be randomly selected lanes for each car. Cars should then be seated after Round 1 so that the top two in each lane for each heat all use the fastest lane in Round 2. The 3-4 finishers in each lane will then race in the second fastest lane. This pattern will continue and the lowest times are then given their awards after the two rounds. A third round would further solidify the results.

[Guest]

I think this is an opportunity for a KISS application

or Keep it Simple Stupid. Lane 4 appears to be the big problem. It sounds like last year you had 40 heats, and 21 wins from lane 4, with six or seven wins in lanes 1 and 2. Just don't use lane 4, and don't allow a car to race in lanes 1 AND 2. Keep the rest of the procedure the same as last year, and you'll be fine.