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Yet Another Race Method

Sure, there are a lot of different types of racing charts out there. So one more wouldn't hurt, would it?
  • Equality of lane assignments - Every car races the same number of times in each lane.
  • Equality of opposition - Every car races the same number of times against every other car.
These are two of the major elements that contribute to the value of a final-standing racing chart.

The extent to which a chart exhibits these two characteristics will correlate directly with the accuracy1 of the chart, that is, the ability of the chart to correctly rank the cars, from fastest to slowest.

It will also correlate directly with the perception of the chart by those competitive parents out there who study the charts and look for flaws. In the other words, a chart should not only be accurate, it should look accurate.

Perfect-N charts are the gemstones of racing charts - somewhat rare, virtually flawless, and often expensive. They fully satisfy both of the conditions above, and because of this, exhibit very strong accuracy. Unfortunately, because of their strict criteria and because of the somewhat complex mathematics of racing charts, Perfect-N charts only exist for certain combinations of cars and lanes. And for large numbers of cars, the charts are usually too large to be useful, even when they do exist. Thus, these charts are primarily useful for small groups, such as Dens, or after a large group has been trimmed to a small number of "finalists". Perfect-N charts are documented in detail by Stan Pope.

Lane Rotation charts are at the other end of the spectrum. They are easy to create, they exist for any combination of cars and lanes, and they even satisfy equality of lane assignments. However, they have very poor equality of opposition, and this shows when accuracy simulations are run against them. Lane Rotation charts are documented in Darin McGrew's excellent essay on race methods.

Somewhere in the middle are Stearns Method charts. They are very flexible - you can generate them for any combination of cars and lanes. And unlike Perfect-N and Lane Rotation, the number of races does not have to equal a multiple of the number of cars, which provides further adaptability. On the other hand, although an attempt is made to satisfy both of the equality conditions above, it is a rare Stearns chart that does so completely. Further, Stearns charts often exhibit peculiarities, such as one car racing some cars three times, and others only once. When certain cars are involved, peculiarities such as this become controversial.

Here is a Lane Rotation chart for 8 cars on 4 lanes. Note the poor equality of opposition. For example, Car 1 has three races with Car 2, but none with Car 5.

         Ln1 Ln2 Ln3 Ln4
Heat 1    1   2   3   4
Heat 2    2   3   4   5
Heat 3    3   4   5   6
Heat 4    4   5   6   7
Heat 5    5   6   7   8
Heat 6    6   7   8   1
Heat 7    7   8   1   2
Heat 8    8   1   2   3
By adding 1 to each of the Car numbers in the first race, you get the Car numbers for the second race. Do the same for the second race and you get the Car numbers for the third race. And so on. (Of course, you have to "wrap around" to Car 1 when you add 1 to Car 8.)

What might happen if, instead of starting with race 1-2-3-4, we started with a different race, say, 1-3-4-7, and then incremented the car numbers to complete the chart?

         Ln1 Ln2 Ln3 Ln4
Heat 1    1   3   4   7
Heat 2    2   4   5   8
Heat 3    3   5   6   1
Heat 4    4   6   7   2
Heat 5    5   7   8   3
Heat 6    6   8   1   4
Heat 7    7   1   2   5
Heat 8    8   2   3   6
Each car still races once in each lane, but now the head-to-head matchups are much improved. Car 1 now races twice each against 5 of the cars, and once each against the other two. And since Car 1 has a total of 12 opponents (4 races times 3 opponents per race), this chart demonstrates best possible equality of opposition, given the dimensions of the chart.

This method of "starting with a good race and incrementing car numbers" to generate a chart is how Stan Pope's Perfect-N charts are created. The criteria for Perfect-N charts is so strict, though, that compliant charts are somewhat rare.

Our new style of chart relaxes the "perfect equality of opposition" condition to "best possible equality of opposition" . So we might expect charts of this type to be much more common.

So we now have a new type of chart, which we'll call Partial Perfect-N (aka PPN, aka Enhanced Lane Rotation, aka Perfect-N wannabe). Such a chart satisfies the conditions that:

  • The chart has perfect equality of lane assignments.
  • The chart has best-possible equality of opposition. 2
PPN charts, it turns out, are much more accurate than ordinary Lane Rotation charts, and, in general, they are even slightly more accurate than Stearns charts.3 They also exhibit none of the potential controversialities that are often found in Stearns charts.

Further, single round (# of races equals # of cars) PPN charts exist for any number of cars on tracks up to and including 4 lanes. For 5 and 6 lane tracks, they exist most of the time, much more often than Perfect-N. Double round PPN charts are only slightly more scarce.

So what are PPN charts good for? Well, for not too large groups (say, less than 60 cars), you could use a PPN chart to run your entire Derby. Even better, use a PPN chart to trim down to a handful of finalists, then use a Perfect-N chart to determine trophies. This composite method does a great job of balancing participation, time constraints, and accurate selection of winners, as documented by Stan Pope on his Case Study page.

So how does one create PPN charts? Stan (who else?) has created a web-based PPN chart generator, somewhat similar to the Perfect-N chart generator which has been available since early 1998. It generates charts in a variety of formats which should suit the needs of most people.

1It is possible to simulate the running of a racing chart over a great many Pinewood Derbies via software. The results can be tallied, and then trended, to determine how well a chart correctly awards trophies to the fastest cars, overcoming obstacles like uneven lanes and random elements. Such a simulation tool can be found on our Software page. Results from this tool are the basis for the accuracy claims made on this page.

2 Another way of stating this second condition is that no head-to-head matchup count between two cars should exceed any other head-to-head matchup count by more than one. For example, suppose Cars 1 and 2 have two races together. Then, Cars 3 and 4 (or any other pair of cars) should have no less than 1 race, and no more than 3 races, together. And if that number is, say, 3, then Cars 5 and 6 (or any other pair of cars) should have either 2 or 3 races together.

3 Here are some comparisons using the simulation tool. 

8 car 4 lane 16 race PPN Chart 

 1-Trophy Accuracy:  88.10%     Top-1 Accuracy:  88.10%
 2-Trophy Accuracy:  79.72%     Top-2 Accuracy:  89.97%
 3-Trophy Accuracy:  74.57%     Top-3 Accuracy:  92.38%     3n3 Accuracy:  92.38%
 4-Trophy Accuracy:  71.35%     Top-4 Accuracy:  93.81%     3n4 Accuracy:  98.45%
 5-Trophy Accuracy:  69.23%     Top-5 Accuracy:  95.31%     3n5 Accuracy:  99.70%
 6-Trophy Accuracy:  68.60%     Top-6 Accuracy:  96.83%     3n6 Accuracy:  99.97%
 7-Trophy Accuracy:  69.12%     Top-7 Accuracy:  98.21%     3n7 Accuracy: 100.00%
 
8 car 4 lane 16 race Stearns Chart

 1-Trophy Accuracy:  85.80%     Top-1 Accuracy:  85.80%
 2-Trophy Accuracy:  77.22%     Top-2 Accuracy:  88.85%
 3-Trophy Accuracy:  71.18%     Top-3 Accuracy:  90.33%     3n3 Accuracy:  90.33%
 4-Trophy Accuracy:  66.78%     Top-4 Accuracy:  92.11%     3n4 Accuracy:  97.73%
 5-Trophy Accuracy:  64.39%     Top-5 Accuracy:  94.53%     3n5 Accuracy:  99.58%
 6-Trophy Accuracy:  63.35%     Top-6 Accuracy:  96.07%     3n6 Accuracy:  99.95%
 7-Trophy Accuracy:  64.17%     Top-7 Accuracy:  98.19%     3n7 Accuracy: 100.00%

16 car 4 lane 32 race PPN chart

 1-Trophy Accuracy:  83.20%     Top-1 Accuracy:  83.20%
 2-Trophy Accuracy:  73.15%     Top-2 Accuracy:  86.13%
 3-Trophy Accuracy:  66.43%     Top-3 Accuracy:  87.48%     3n3 Accuracy:  87.48%
 4-Trophy Accuracy:  60.91%     Top-4 Accuracy:  88.74%     3n4 Accuracy:  95.72%
 5-Trophy Accuracy:  56.82%     Top-5 Accuracy:  90.65%     3n5 Accuracy:  98.62%
 6-Trophy Accuracy:  53.51%     Top-6 Accuracy:  91.22%     3n6 Accuracy:  99.52%
 7-Trophy Accuracy:  51.24%     Top-7 Accuracy:  92.59%     3n7 Accuracy:  99.78%

16 car 4 lane 32 race Stearns chart

 1-Trophy Accuracy:  80.60%     Top-1 Accuracy:  80.60%
 2-Trophy Accuracy:  70.30%     Top-2 Accuracy:  84.35%
 3-Trophy Accuracy:  62.50%     Top-3 Accuracy:  86.17%     3n3 Accuracy:  86.17%
 4-Trophy Accuracy:  57.49%     Top-4 Accuracy:  88.30%     3n4 Accuracy:  94.80%
 5-Trophy Accuracy:  53.67%     Top-5 Accuracy:  89.68%     3n5 Accuracy:  98.23%
 6-Trophy Accuracy:  50.40%     Top-6 Accuracy:  90.64%     3n6 Accuracy:  99.47%
 7-Trophy Accuracy:  47.99%     Top-7 Accuracy:  91.80%     3n7 Accuracy:  99.78%

32 car 4 lane 32 race PPN Chart

 1-Trophy Accuracy:  77.25%     Top-1 Accuracy:  77.25%
 2-Trophy Accuracy:  61.28%     Top-2 Accuracy:  71.90%
 3-Trophy Accuracy:  50.35%     Top-3 Accuracy:  75.42%     3n3 Accuracy:  75.42%
 4-Trophy Accuracy:  43.89%     Top-4 Accuracy:  77.05%     3n4 Accuracy:  85.10%
 5-Trophy Accuracy:  38.84%     Top-5 Accuracy:  79.34%     3n5 Accuracy:  91.00%
 6-Trophy Accuracy:  35.34%     Top-6 Accuracy:  81.27%     3n6 Accuracy:  94.48%
 7-Trophy Accuracy:  32.46%     Top-7 Accuracy:  82.31%     3n7 Accuracy:  96.28%

32 car 4 lane 32 race Stearns Chart

 1-Trophy Accuracy:  79.10%     Top-1 Accuracy:  79.10%
 2-Trophy Accuracy:  64.15%     Top-2 Accuracy:  73.83%
 3-Trophy Accuracy:  52.50%     Top-3 Accuracy:  75.00%     3n3 Accuracy:  75.00%
 4-Trophy Accuracy:  44.96%     Top-4 Accuracy:  76.96%     3n4 Accuracy:  85.10%
 5-Trophy Accuracy:  39.53%     Top-5 Accuracy:  78.40%     3n5 Accuracy:  90.88%
 6-Trophy Accuracy:  35.81%     Top-6 Accuracy:  79.90%     3n6 Accuracy:  94.18%
 7-Trophy Accuracy:  32.74%     Top-7 Accuracy:  81.12%     3n7 Accuracy:  96.40%

48 car 4 lane 48 race PPN Chart

 1-Trophy Accuracy:  80.35%     Top-1 Accuracy:  80.35%
 2-Trophy Accuracy:  63.73%     Top-2 Accuracy:  71.22%
 3-Trophy Accuracy:  50.92%     Top-3 Accuracy:  70.37%     3n3 Accuracy:  70.37%
 4-Trophy Accuracy:  43.16%     Top-4 Accuracy:  74.09%     3n4 Accuracy:  80.78%
 5-Trophy Accuracy:  38.48%     Top-5 Accuracy:  76.38%     3n5 Accuracy:  86.85%
 6-Trophy Accuracy:  34.59%     Top-6 Accuracy:  76.83%     3n6 Accuracy:  91.07%
 7-Trophy Accuracy:  31.39%     Top-7 Accuracy:  77.90%     3n7 Accuracy:  93.67%

48 car 4 lane 48 race Stearns Chart

 1-Trophy Accuracy:  81.50%     Top-1 Accuracy:  81.50%
 2-Trophy Accuracy:  64.78%     Top-2 Accuracy:  71.47%
 3-Trophy Accuracy:  51.73%     Top-3 Accuracy:  69.93%     3n3 Accuracy:  69.93%
 4-Trophy Accuracy:  43.25%     Top-4 Accuracy:  72.50%     3n4 Accuracy:  80.43%
 5-Trophy Accuracy:  37.91%     Top-5 Accuracy:  74.50%     3n5 Accuracy:  86.73%
 6-Trophy Accuracy:  33.80%     Top-6 Accuracy:  75.38%     3n6 Accuracy:  90.83%
 7-Trophy Accuracy:  30.53%     Top-7 Accuracy:  76.73%     3n7 Accuracy:  93.57%
Acknowledgement is due to Stan Pope for the wisdom, experience, and creativity he brought to this page. Besides coining the term "Partial Perfect-N", he also provided the scheme by which I was able to write a generator-finder for PPN charts, the results of which are used in Stan's JavaScript Partial Perfect-N Chart Generator.

Last updated on December 13, 1998, 2:00 PM
Copyright 1998 © by Cory Young. All rights reserved.