Sports League Scheduling - Using an Imaginary Bead Necklace

My friend Warren has to schedule a 13-week, 13-Team Billiards League and could not figure out how to do it, so he asked me for help. OY!

After spending an inordinate amount of time on and off over three days, at around 4AM this morning, I finally came up with an almost embarrassingly simple way to visualize the problem and solve it!

My solution is shown above using a spreadsheet format. Each column and each row represent a given Team, and each column/row intersection represents a given Team pairing, on a specified Week and at a specified Table. Each of the 13 weeks is shown in a different color. Since 13 is an ODD number of Teams, each week one Team has to take a "BYE" and not play. The main diagonal indicates that Team #1 will take a BYE on Week 1, Team #2 on Week 2, and so on.

AN IMAGINARY BEAD NECKLACE PROVIDES THE SOLUTION!

The diagram below shows how you can utilize an imaginary bead necklace to solve any Sports League Scheduling problem, including both Leagues with an ODD number of Teams and Leagues with an EVEN number of Teams.

The blue necklace represents a case where there are an ODD number of Teams, in this case 13. Imagine you are holding one of the beads (in this case bead "5") and allowing the other beads to stretch the necklace downwards. This represents one week of League play, say Week 5. The "5" bead you are holding is the Team that will take a BYE on that week. The next two beads ("3" and "4") indicate that Teams #3 and #4 will be paired on that week. Looking down the necklace, Teams #2 and #6 will also be paired on Week 5, as will Teams #1 and #7, #13 and #8, #12 and #9, and #11 and #10. Thus, we will need six tables to accommodate the six pairs of Teams.

To get the pairings for any other week, hold the corresponding numbered bead and read the pairings!

The pink necklace represents a case where there are an EVEN number of Teams, in this case 14. Imagine you are holding the STRING SEGMENT between two of the beads (in this case between bead "5" and bead "6") and allowing the other beads to stretch the necklace downwards. This represents one week of League play, say Week 6. Since there are an EVEN number of Teams, no one will have to take a BYE on that week. The top two beads ("5" and "6") indicate that Teams #5 and #6 will be paired on that week. Looking down the necklace, Teams #4 and #7 will also be paired on Week 6, as will Teams #3 and #8, #2 and #9, #1 and #10, #14 and #11, and #13 and #12. Thus, we will need seven tables to accommodate the seven pairs of Teams.

To get the pairings for any other week, hold the STRING SEGMENT that links the corresponding numbered bead to its predecessor and read the pairings!

CONCLUSION

I hope this will help anyone who needs to schedule a Sports League with any number of Teams. Please post a comment if you find this helpful (or if you have any suggestions on how I can make it clearer.)

Ira Glickstein

Creativity

I'm thinking of giving a talk to our local philosophy club. The topic will be creativity and imagination. The subtopics will be as follows. What is your definition of creativity? Is creativity inborn or learned, or a mixture of the two? If it's at least partly learned, can it be taught or enhanced? What part does visualization (or imagination) play in creativity? Can visualization be taught?

Historically there are three sources of creativity that we might consider. For the Greeks creativity was a mystical process that required the intervention of a supernatural Muse. David Hume and John Locke thought that creativity was nothing more than the ability to rearrange previous ideas or perceptions. Others see creativity as the fortunate application of pure chance.

I taught what might be called "psuedo creativity" for a few years. If we use the Webster's Encyclopedic Dictionary definition for creativity, it would be the ability to transcend traditional rules, ideas, patterns, etc. in order to generate meaningful new ideas. Pseudo creativity is the application of methods that will generate unnatural thinking paths that will generate new ideas. For example, most people have unconscious censors in their minds that will eliminate thought paths that are illegal, immoral or violate the laws of physics. This constraining of the the process at a very early stage leads to narrow thinking. If we consciously eliminate such constraints we may come up with ideas or problem solutions that are not physically or legally or morally realizable, but they may have distant cousins that are achievable.

I would appreciate your ideas on the subject. With respect-Joel

Magical Technology and the Supernatural

In an earlier thread, Howard introduced a new topic idea: " ... almost no one who uses the Internet has the faintest idea of the physics, engineering, manufacturing and programming skills that are essential for making it work. I mean, it’s a total mystery to most people. "

I agreed and quoted what turned out to be one of Arthur C. Clarke's Laws:

Any sufficiently advanced technology is indistinguishable from magic.

Howard went on: "... this feeling of mystery (or ignorance) may account for the rise the distrust of science in general and the increase in accepting occult and supernatural beliefs. In other words, if technical devices appear to have an unknown cause, then any event can have an unknown cause."

I agree with Howard that technology is a mystery to most people using it, but I am not sure if he is right about a rise in the distrust of science or an increase in occult and supernatural beliefs. In fact, I am concerned the common folks place too much trust in science and, as a result, are losing their faith in the supernatural.

My reading of history is that irrational beliefs are central to the success of any society. If they get out of hand -OR- if they are extinguished by pure reason, disaster follows. As always, the "happy medium" is the path to success.

The scientific method is a totally rational ideal that is largely responsible for modern civilization. However necessary, it is by no means sufficient. We still need belief in supernatural magic and other things not literally true to make progress work.

Another of Clarke's laws states:

The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

Indeed, we must creatively imagine we believe in the impossible to assure progress in extending the limits of science and technology. Howard agrees to a point: "An essential part of the scientific method is entirely rational, but another essential part is creative imagination often using irrational analogies and metaphors. Also, for physicists the natural world is super enough so that the religious supernatural appears quite dull."

We need to appreciate another of Clarke's laws:

When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

I, and most of the other active members of this Blog may be characterized, in the words of this law, as "distinguished but elderly scientists". We need to be wary of declaring anything impossible!

On the other hand, according to Isaac Asimov's corollary to Clarke's law:

"When, however, the lay public rallies round an idea that is denounced by distinguished but elderly scientists and supports that idea with great fervor and emotion -- the distinguished but elderly scientists are then, after all, probably right."

Ira Glickstein

PS: The photograph is of The_Turk, a 1770's chess playing machine. It turned out to be a hoax - a skilled human chess player was hidden inside the device. Today, however, you can buy a chess program that will run on your PC and beat everyone but a chess master. Larger computers are well-matched with human chess masters and, in the forseeable future, the world chess champion will be a computer.