V 12 N. 31 A Response to Scoring Dual Meets With 5-3-1 or 3-2-1 Systems

 I'm sure this response from Gary Andrus on scoring dual track meets with 5-3-1 or 3-2-1 will enlighten many of our readers and stir up the cobwebs in the brains of the rest.   Ned, do you remember Gary?  Did he pass your course?  I'm reading this as a student who squeezed through my only undergrad math class with a D.  However on the broader spectrum that same semester I also got a D in ROTC, although I did learn to use a compass and read a map.  But militarily I  placed my machine guns in the wrong positions.    George

Hi George, Gary Andrus here. 

Love the site, thanks for your fine work.

Back in the late 60's/early 70's Ned Price was a professor in the

math department at Wayne State University in Detroit. At that

time I was a Ph.D. student in math at Wayne. Ned and I frequently

went on training runs together. I can still smell the fumes from

the beer distillery we often ran by on our way down to Cobo Hall.

Anyway, regarding your recent post in which Ned gives a case of

a duel meet where one team loses using a 5-3-1 system of 

scoring, but wins using a 3-2-1 system. To expand on Ned's result,

(given 10 events, no relays) there are a total of 10 cases in which

that phenomenon exists. I wrote a simple Java program to 

compute the cases.

x = number of first places for Team A

y = number of second places for Team A

z = number of third places for Team A

x = 1 y = 9 z = 10

Scoring under 5-3-1

Team A: 42

Team B: 48

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 1 y = 10 z = 8

Scoring under 5-3-1

Team A: 43

Team B: 47

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 1 y = 10 z = 9

Scoring under 5-3-1

Team A: 44

Team B: 46

Team B wins

Scoring under 3-2-1

Team A: 32

Team B: 28

Team A wins

x = 2 y = 8 z = 9

Scoring under 5-3-1

Team A: 43

Team B: 47

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 2 y = 8 z = 10

Scoring under 5-3-1

Team A: 44

Team B: 46

Team B wins

Scoring under 3-2-1

Team A: 32

Team B: 28

Team A wins

x = 2 y = 9 z = 7

Scoring under 5-3-1

Team A: 44

Team B: 46

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 3 y = 6 z = 10

Scoring under 5-3-1

Team A: 43

Team B: 47

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 3 y = 7 z = 8

Scoring under 5-3-1

Team A: 44

Team B: 46

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 4 y = 5 z = 9

Scoring under 5-3-1

Team A: 44

Team B: 46

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

x = 5 y = 3 z = 10

Scoring under 5-3-1

Team A: 44

Team B: 46

Team B wins

Scoring under 3-2-1

Team A: 31

Team B: 29

Team A wins

Total cases = 10

I might as well throw in my source code too. It's not the most

efficient algorithm, but the numbers are small so, execution-wise,

it doesn't make much difference. Also, one can imagine many

generalizations. Maybe your readers would like to work on some.

/**

 * A dual track meet between Team A and Team B. 10 events.

 * x = number of first places for Team A

 * y = number of second places for Team A

 * z = number of third places for Team A

 *

 * Gary Andrus

 * Version 1.0, Date 4/14/2022

 **/

public class TrackScore

{

    /**

     * Output: All possibilities with Team B winning under the 5-3-1

     *         system of scoring but losing under the 3-2-1 system.

     **/

    public static void main (String[] args)

    {

        // points for first, second, and third for two scoring methods

        int f1 = 5, s1 = 3, t1 = 1;

        int f2 = 3, s2 = 2, t2 = 1;

        int x, y, z;

        int count = 0;

       

        for (x=0; x<=8; x++)

            for (y=0; y<=10;y++)

                for (z=0; z<=10; z++)

                {

                    int a = f1*x + s1*y + t1*z;  // Team A points under 5-3-1

                    int b = 90 - a;              // Team B points under 5-3-1

                    int aa = f2*x + s2*y + t2*z; // Team A points under 3-2-1

                    int bb = 60 - aa;            // Team B points under 3-2-1

                    if (a < 45)

                    {

                        if (aa > 30)

                        {

                            System.out.println ();

                            System.out.println ("x = " + x + " y = " + y + " z = " + z);

                            System.out.println

                            ("Scoring under " + f1 +"-" + s1 +"-" + t1);

                            System.out.println ("Team A: " + a);

                            System.out.println ("Team B: " + b);

                            System.out.println ("Team B wins");

                            System.out.println ();

                            System.out.println

                            ("Scoring under " + f2 +"-" + s2 +"-" + t2);

                            System.out.println ("Team A: " + aa);

                            System.out.println ("Team B: " + bb);

                            System.out.println ("Team A wins");

                            count++;

                        }

                    }

                }

        System.out.println ();

        System.out.println ("Total cases = " + count);                

    }

}