"Perfect N" Race Grid Directory

General Information

These racer assignment grids satisfy the "Perfect-N Criteria" in which each car races in each lane the same number of times and races each opponent the same number of times. Competitions which satisfy the "Perfect-N Criteria" have exceptional accuracy because almost all of the effects of lane differences are balanced out.

Two types of "chart symbols" are used: "P N-L (M)" and "CP N-L (M)". "P" indicates that the chart satisfies the "Perfect-N" criteria; "CP" indicates that the chart satisfies the more accurate "Complementary Perfect-N" criteria. N is the number of cars, L is the number of lanes, and M is the number of times each pair of cars is matched.

This page provides the parameters for quite a few charts (about 90 of them) for a variety of track and group sizes. If you have a JavaScript-capable browser, such as Netscape or MS Internet Explorer, your browser can prepare the charts for you. Try the Perfect-N Chart Generation page.

Recommended Scoring:

Assign points according to finish place in each heat. Total each car's points at the end.

Selection:

Select grids which requires no more lanes than you have available. From those, select charts which can handle the number of cars you have to race. From those, select a chart which has an appropriate number of heats.

Note: The method retains significant accuracy, even if byes are raced.

Chart Generation:

Steps:

(see example)

Note N, the number of cars; H, the number of heats; L, the number of lanes; and G(i), the generators for the selected chart.
Draw a grid of L columns and H rows.
Enter the numbers 1, 2, 3, ... down column 1. If a result is greater than N, subtract N from it. Continue until column 1 is full. (The number in the last row should be N. If it isn't then there has been an error.)
Next to each row that has a "1" in column 1, write one of the generators. Each generator should be used once. If not, then there has been an error.
In the following addition, if a result is greater than N, subtract N from it to produce a number in the range 1 thru N.

Apply each generator to fill in the remainder of it's row: Add the first number in the gernerator to the contents of the first column to produce a value for the second column. Add that number to the second number in the generator to produce a value for the third column. Continue in that fashion until all of the generator values have been used and each of the columns of the row have been filled. (It should come out even...if not, there has been an error.)
Fill in the remainder of each column by numbering sequentially down the column, beginning with the values already present. The next number after N is "1". Here again, if the number you are about to enter is larger than N, subtract N from it.

Directory:

Cars Heats Chart Symbol Generator(s)

3 Lanes

3 3 P 3-3 (3) {1, 1}

3 6 CP 3-3 (6) {1, 1}
{2, 2}

4 4 P4-3 (2) {1, 1}

4 8 CP4-3 (4) {1, 1}
{3, 3}

5 10 P5-3 (3) {1, 1}
{2, 2}

7 7 P7-3 (1) {1, 2}

7 14 P7-3 (2) {1, 2}
{2, 1}

7 14 CP7-3 (2) {1, 2}
{6, 5}

7 21 P7-3 (3) {1, 1}
{2, 2}
{3, 3}

7 28 CP7-3 (4) {1, 3}
{6, 4}
{3, 2}
{4, 5}

9 36 P9-3 (3) {1, 1}
{2, 2}
{3, 3}
{4, 4}

9 72 CP9-3 (6) {1, 1}
{8, 8}
{2, 2}
{7, 7}
{3, 3}
{6, 6}
{4, 4}
{5, 5}

13 26 P13-3 (1) {1, 3}
{2, 5}

13 52 CP13-3 (2) {1, 3}
{12, 10}
{2, 5}
{11, 8}

19 57 P19-3 (1) {1, 3}
{2, 7}
{5, 6}

19 114 CP19-3 (2) {1, 3}
{18, 16}
{2, 7}
{17, 12}
{5, 6}
{14, 13}

25 100 P25-3 (1) {1, 2}
{4, 7}
{5, 8}
{6, 9}

25 200 CP25-3 (2) {1, 2}
{24, 23}
{4, 7}
{21, 18}
{5, 8}
{20, 17}
{6, 9}
{19, 16}

4 Lanes

4 4 P4-4 (4) {1, 1, 1}

4 8 CP4-4 (8) {1, 1, 1}
{3, 3, 3}

5 5 P5-4 (3) {1, 1, 1}

5 10 CP5-4 (6) {1, 1, 1}
{4, 4, 4}

5 15 P5-4 (9) {1, 1, 1}
{2, 2, 2}
{3, 3, 3}

7 7 P7-4 (2) {1, 1, 2}

7 14 CP7-4 (4) {1, 1, 2}
{6, 6, 5}

7 21 P7-4 (6) {1, 1, 1}
{2, 2, 2}
{3, 3, 3}

9 18 P9-4 (3) {1, 1, 2}
{2, 3, 1}

9 36 CP9-4 (6) {1, 1, 2}
{8, 8, 7}
{2, 3, 1}
{7, 6, 8}

10 30 P10-4 (4) {1, 1, 2}
{2, 2, 3}
{3, 6, 5}

10 60 CP10-4 (8) {1, 1, 2}
{9, 9, 8}
{2, 2, 3}
{8, 8, 7}
{3, 6, 5}
{7, 4, 5}

13 13 P13-4 (1) {1, 2, 6}

13 26 CP13-4 (2) {1, 2, 6}
{12, 11, 7}

13 39 P13-4 (3) {1, 1, 3}
{2, 5, 5}
{3, 4, 2}

19 57 P19-4 (2) {1, 1, 4}
{2, 6, 3}
{3, 4, 5}

19 114 CP19-4 (4) {1, 1, 4}
{18, 18, 15}
{2, 6, 3}
{17, 13, 16}
{3, 4, 5}
{16, 15, 14}

37 111 P37-4 (1) {1, 2, 21}
{4, 5, 6}
{7, 10, 8}

37 222 CP37-4 (2) {1, 2, 21}
{36, 35, 16}
{4, 5, 6}
{33, 32, 31}
{7, 10, 8}
{30, 27, 29}

5 Lanes

5 5 P5-5 (5) {1, 1, 1, 1}

5 10 CP5-5 (10) {1, 1, 1, 1}
{4, 4, 4, 4}

6 6 P6-5 (4) {1, 1, 1, 1}

6 12 CP6-5 (8) {1, 1, 1, 1}
{5, 5, 5, 5}

6 18 P6-5 (12) {1, 1, 1, 1}
{2, 2, 3, 2}
{3, 4, 4, 3}

7 21 P7-5 (10) {1, 1, 1, 1}
{2, 2, 2, 2}
{3, 3, 3, 3}

7 42 CP7-5 (20) {1, 1, 1, 1}
{6, 6, 6, 6}
{2, 2, 2, 2}
{5, 5, 5, 5}
{3, 3, 3, 3}
{4, 4, 4, 4}

9 18 P9-5 (5) {1, 1, 1, 3}
{2, 2, 3, 1}

9 36 CP9-5 (10) {1, 1, 1, 3}
{8, 8, 8, 6}
{2, 2, 3, 1}
{7, 7, 6, 8}

11 11 P11-5 (2) {1, 1, 2, 3}

11 22 CP11-5 (4) {1, 1, 2, 3}
{10, 10, 9, 8}

21 21 P21-5 (1) {1, 3, 10, 2}

21 42 CP21-5 (2) {1, 3, 10, 2}
{20, 18, 11, 19}

41 82 P41-5 (1) {1, 3, 7, 18}
{2, 6, 9, 5}

41 164 CP41-5 (2) {1, 3, 7, 18}
{40, 38, 34, 23}
{2, 6, 9, 5}
{39, 35, 32, 36}

6 Lanes

6 6 P6-6 (6) {1, 1, 1, 1, 1}

6 12 CP6-6 (12) {1, 1, 1, 1, 1}
{5, 5, 5, 5, 5}

7 7 P7-6 (5) {1, 1, 1, 1, 1}

7 14 CP7-6 (10) {1, 1, 1, 1, 1}
{6, 6, 6, 6, 6}

7 21 P7-6 (15) {1, 1, 1, 1, 1}
{2, 2, 2, 2, 2}
{3, 3, 3, 3, 3}

11 11 P11-6 (3) {1, 1, 2, 1, 2}

11 22 CP11-6 (6) {1, 1, 2, 1, 2}
{10, 10, 9, 10, 9}

13 26 P13-6 (5) {1, 1, 1, 2, 4}
{2, 3, 3, 1, 7}

13 52 CP13-6 (10) {1, 1, 1, 2, 4}
{12, 12, 12, 11, 9}
{2, 3, 3, 1, 7}
{11, 10, 10, 12, 6}

31 31 P31-6 (1) {1, 2, 5, 4, 6}

31 62 CP31-6 (2) {1, 2, 5, 4, 6}
{30, 29, 26, 27, 25}

7 Lanes

7 7 P7-7 (7) {1, 1, 1, 1, 1, 1}

7 14 CP7-7 (14) {1, 1, 1, 1, 1, 1}
{6, 6, 6, 6, 6, 6}

8 8 P8-7 (6) {1, 1, 1, 1, 1, 1}

8 16 CP8-7 (12) {1, 1, 1, 1, 1, 1}
{7, 7, 7, 7, 7, 7}

15 15 P15-7 (3) {1, 1, 2, 1, 3, 2}

15 30 CP15-7 (6) {1, 1, 2, 1, 3, 2}
{14, 14, 13, 14, 12, 13}

8 Lanes

8 8 P8-8 (8) {1, 1, 1, 1, 1, 1, 1}

8 16 CP8-8 (16) {1, 1, 1, 1, 1, 1, 1}
{7, 7, 7, 7, 7, 7, 7}

9 9 P9-8 (7) {1, 1, 1, 1, 1, 1, 1}

9 18 CP9-8 (14) {1, 1, 1, 1, 1, 1, 1}
{8, 8, 8, 8, 8, 8, 8}

15 15 P15-8 (4) {1, 1, 1, 2, 2, 1, 3}

15 30 CP15-8 (8) {1, 1, 1, 2, 2, 1, 3}
{14, 14, 14, 13, 13, 14, 12}

57 57 P57-8 (1) {1, 2, 10, 19, 4, 7, 9}

57 114 CP57-8 (2) {1, 2, 10, 19, 4, 7, 9}
{56, 55, 47, 38, 53, 50, 48}

9 Lanes

9 9 P9-9 (9) {1, 1, 1, 1, 1, 1, 1, 1}

9 18 CP9-9 (18) {1, 1, 1, 1, 1, 1, 1, 1}
{8, 8, 8, 8, 8, 8, 8, 8}

10 10 P10-9 (8) {1, 1, 1, 1, 1, 1, 1, 1}

10 20 CP10-9 (16) {1, 1, 1, 1, 1, 1, 1, 1}
{9, 9, 9, 9, 9, 9, 9, 9}

13 13 P13-9 (6) {1, 1, 1, 1, 1, 2, 2, 1}

13 26 CP13-9 (12) {1, 1, 1, 1, 1, 2, 2, 1}
{12, 12, 12, 12, 12, 11, 11, 12}

19 19 P19-9 (4) {1, 1, 1, 2, 2, 5, 1, 3}

19 38 CP19-9 (8) {1, 1, 1, 2, 2, 5, 1, 3}
{18, 18, 18, 17, 17, 14, 18, 16}

37 37 P37-9 (2) {1, 2, 4, 10, 7, 1, 4, 6}

37 74 CP37-9 (4) {1, 2, 4, 10, 7, 1, 4, 6}
{36, 35, 33, 27, 30, 36, 33, 31}

73 73 P73-9 (1) {1, 2, 4, 6, 16, 5, 18, 9}

73 146 CP73-9 (2) {1, 2, 4, 6, 16, 5, 18, 9}
{72, 71, 69, 67, 57, 68, 55, 64}

10 Lanes

91 91 P91-10 (1) {1, 2, 6, 18, 22, 7, 5, 16, 4}

91 182 CP91-10 (2) {1, 2, 6, 18, 22, 7, 5, 16, 4}
{90, 89, 85, 73, 69, 84, 86, 75, 87}

Cars	Heats	Chart Symbol	Generator(s)
3 Lanes
3	3	P 3-3 (3)	{1, 1}
3	6	CP 3-3 (6)	{1, 1} {2, 2}
4	4	P4-3 (2)	{1, 1}
4	8	CP4-3 (4)	{1, 1} {3, 3}
5	10	P5-3 (3)	{1, 1} {2, 2}
7	7	P7-3 (1)	{1, 2}
7	14	P7-3 (2)	{1, 2} {2, 1}
7	14	CP7-3 (2)	{1, 2} {6, 5}
7	21	P7-3 (3)	{1, 1} {2, 2} {3, 3}
7	28	CP7-3 (4)	{1, 3} {6, 4} {3, 2} {4, 5}
9	36	P9-3 (3)	{1, 1} {2, 2} {3, 3} {4, 4}
9	72	CP9-3 (6)	{1, 1} {8, 8} {2, 2} {7, 7} {3, 3} {6, 6} {4, 4} {5, 5}
13	26	P13-3 (1)	{1, 3} {2, 5}
13	52	CP13-3 (2)	{1, 3} {12, 10} {2, 5} {11, 8}
19	57	P19-3 (1)	{1, 3} {2, 7} {5, 6}
19	114	CP19-3 (2)	{1, 3} {18, 16} {2, 7} {17, 12} {5, 6} {14, 13}
25	100	P25-3 (1)	{1, 2} {4, 7} {5, 8} {6, 9}
25	200	CP25-3 (2)	{1, 2} {24, 23} {4, 7} {21, 18} {5, 8} {20, 17} {6, 9} {19, 16}
4 Lanes
4	4	P4-4 (4)	{1, 1, 1}
4	8	CP4-4 (8)	{1, 1, 1} {3, 3, 3}
5	5	P5-4 (3)	{1, 1, 1}
5	10	CP5-4 (6)	{1, 1, 1} {4, 4, 4}
5	15	P5-4 (9)	{1, 1, 1} {2, 2, 2} {3, 3, 3}
7	7	P7-4 (2)	{1, 1, 2}
7	14	CP7-4 (4)	{1, 1, 2} {6, 6, 5}
7	21	P7-4 (6)	{1, 1, 1} {2, 2, 2} {3, 3, 3}
9	18	P9-4 (3)	{1, 1, 2} {2, 3, 1}
9	36	CP9-4 (6)	{1, 1, 2} {8, 8, 7} {2, 3, 1} {7, 6, 8}
10	30	P10-4 (4)	{1, 1, 2} {2, 2, 3} {3, 6, 5}
10	60	CP10-4 (8)	{1, 1, 2} {9, 9, 8} {2, 2, 3} {8, 8, 7} {3, 6, 5} {7, 4, 5}
13	13	P13-4 (1)	{1, 2, 6}
13	26	CP13-4 (2)	{1, 2, 6} {12, 11, 7}
13	39	P13-4 (3)	{1, 1, 3} {2, 5, 5} {3, 4, 2}
19	57	P19-4 (2)	{1, 1, 4} {2, 6, 3} {3, 4, 5}
19	114	CP19-4 (4)	{1, 1, 4} {18, 18, 15} {2, 6, 3} {17, 13, 16} {3, 4, 5} {16, 15, 14}
37	111	P37-4 (1)	{1, 2, 21} {4, 5, 6} {7, 10, 8}
37	222	CP37-4 (2)	{1, 2, 21} {36, 35, 16} {4, 5, 6} {33, 32, 31} {7, 10, 8} {30, 27, 29}
5 Lanes
5	5	P5-5 (5)	{1, 1, 1, 1}
5	10	CP5-5 (10)	{1, 1, 1, 1} {4, 4, 4, 4}
6	6	P6-5 (4)	{1, 1, 1, 1}
6	12	CP6-5 (8)	{1, 1, 1, 1} {5, 5, 5, 5}
6	18	P6-5 (12)	{1, 1, 1, 1} {2, 2, 3, 2} {3, 4, 4, 3}
7	21	P7-5 (10)	{1, 1, 1, 1} {2, 2, 2, 2} {3, 3, 3, 3}
7	42	CP7-5 (20)	{1, 1, 1, 1} {6, 6, 6, 6} {2, 2, 2, 2} {5, 5, 5, 5} {3, 3, 3, 3} {4, 4, 4, 4}
9	18	P9-5 (5)	{1, 1, 1, 3} {2, 2, 3, 1}
9	36	CP9-5 (10)	{1, 1, 1, 3} {8, 8, 8, 6} {2, 2, 3, 1} {7, 7, 6, 8}
11	11	P11-5 (2)	{1, 1, 2, 3}
11	22	CP11-5 (4)	{1, 1, 2, 3} {10, 10, 9, 8}
21	21	P21-5 (1)	{1, 3, 10, 2}
21	42	CP21-5 (2)	{1, 3, 10, 2} {20, 18, 11, 19}
41	82	P41-5 (1)	{1, 3, 7, 18} {2, 6, 9, 5}
41	164	CP41-5 (2)	{1, 3, 7, 18} {40, 38, 34, 23} {2, 6, 9, 5} {39, 35, 32, 36}
6 Lanes
6	6	P6-6 (6)	{1, 1, 1, 1, 1}
6	12	CP6-6 (12)	{1, 1, 1, 1, 1} {5, 5, 5, 5, 5}
7	7	P7-6 (5)	{1, 1, 1, 1, 1}
7	14	CP7-6 (10)	{1, 1, 1, 1, 1} {6, 6, 6, 6, 6}
7	21	P7-6 (15)	{1, 1, 1, 1, 1} {2, 2, 2, 2, 2} {3, 3, 3, 3, 3}
11	11	P11-6 (3)	{1, 1, 2, 1, 2}
11	22	CP11-6 (6)	{1, 1, 2, 1, 2} {10, 10, 9, 10, 9}
13	26	P13-6 (5)	{1, 1, 1, 2, 4} {2, 3, 3, 1, 7}
13	52	CP13-6 (10)	{1, 1, 1, 2, 4} {12, 12, 12, 11, 9} {2, 3, 3, 1, 7} {11, 10, 10, 12, 6}
31	31	P31-6 (1)	{1, 2, 5, 4, 6}
31	62	CP31-6 (2)	{1, 2, 5, 4, 6} {30, 29, 26, 27, 25}
7 Lanes
7	7	P7-7 (7)	{1, 1, 1, 1, 1, 1}
7	14	CP7-7 (14)	{1, 1, 1, 1, 1, 1} {6, 6, 6, 6, 6, 6}
8	8	P8-7 (6)	{1, 1, 1, 1, 1, 1}
8	16	CP8-7 (12)	{1, 1, 1, 1, 1, 1} {7, 7, 7, 7, 7, 7}
15	15	P15-7 (3)	{1, 1, 2, 1, 3, 2}
15	30	CP15-7 (6)	{1, 1, 2, 1, 3, 2} {14, 14, 13, 14, 12, 13}
8 Lanes
8	8	P8-8 (8)	{1, 1, 1, 1, 1, 1, 1}
8	16	CP8-8 (16)	{1, 1, 1, 1, 1, 1, 1} {7, 7, 7, 7, 7, 7, 7}
9	9	P9-8 (7)	{1, 1, 1, 1, 1, 1, 1}
9	18	CP9-8 (14)	{1, 1, 1, 1, 1, 1, 1} {8, 8, 8, 8, 8, 8, 8}
15	15	P15-8 (4)	{1, 1, 1, 2, 2, 1, 3}
15	30	CP15-8 (8)	{1, 1, 1, 2, 2, 1, 3} {14, 14, 14, 13, 13, 14, 12}
57	57	P57-8 (1)	{1, 2, 10, 19, 4, 7, 9}
57	114	CP57-8 (2)	{1, 2, 10, 19, 4, 7, 9} {56, 55, 47, 38, 53, 50, 48}
9 Lanes
9	9	P9-9 (9)	{1, 1, 1, 1, 1, 1, 1, 1}
9	18	CP9-9 (18)	{1, 1, 1, 1, 1, 1, 1, 1} {8, 8, 8, 8, 8, 8, 8, 8}
10	10	P10-9 (8)	{1, 1, 1, 1, 1, 1, 1, 1}
10	20	CP10-9 (16)	{1, 1, 1, 1, 1, 1, 1, 1} {9, 9, 9, 9, 9, 9, 9, 9}
13	13	P13-9 (6)	{1, 1, 1, 1, 1, 2, 2, 1}
13	26	CP13-9 (12)	{1, 1, 1, 1, 1, 2, 2, 1} {12, 12, 12, 12, 12, 11, 11, 12}
19	19	P19-9 (4)	{1, 1, 1, 2, 2, 5, 1, 3}
19	38	CP19-9 (8)	{1, 1, 1, 2, 2, 5, 1, 3} {18, 18, 18, 17, 17, 14, 18, 16}
37	37	P37-9 (2)	{1, 2, 4, 10, 7, 1, 4, 6}
37	74	CP37-9 (4)	{1, 2, 4, 10, 7, 1, 4, 6} {36, 35, 33, 27, 30, 36, 33, 31}
73	73	P73-9 (1)	{1, 2, 4, 6, 16, 5, 18, 9}
73	146	CP73-9 (2)	{1, 2, 4, 6, 16, 5, 18, 9} {72, 71, 69, 67, 57, 68, 55, 64}
10 Lanes
91	91	P91-10 (1)	{1, 2, 6, 18, 22, 7, 5, 16, 4}
91	182	CP91-10 (2)	{1, 2, 6, 18, 22, 7, 5, 16, 4} {90, 89, 85, 73, 69, 84, 86, 75, 87}