8 Races Between 16 Cars Question

resol · 2009-03-03 11:08:01

Ive used all the brain power I have and its not close to enough.

Im having a pinewood derby race with 16 cars and 8 races. Ive figured out how to get everyone to race 4 out of 8 times. My problem is that I want everyone to get to race against everyone else at least once but when I try to move a car it affects more than I can keep up with.

Is there a table I can punch these numbers into so that it will cross-reference with all the other numbers?

Heres what I have so far:

Race - 1 2 3 4 5 6 7 8
_____________________________

Car - 1 9 15 2 5 1 1 2
2 10 13 4 6 2 6 3
3 11 11 6 7 3 7 4
4 12 9 8 8 4 8 5
5 13 7 10 12 9 12 9
6 14 5 12 13 10 13 10
7 15 3 14 14 11 14 11
8 16 1 16 15 16 15 16

Thanks, Derek

mathsyperson · 2009-03-03 11:27:36

Here's one way to do it:

Arrange the cars into four groups of 4 (call the groups A, B, C, D).
Then make 6 of the races like this:

A+B
A+C
A+D
B+C
B+D
C+D

At this point, every car has raced 3 times and every pair of cars has been in the same race at least once. To finish, just put each of the 16 cars into one of the remaining slots in the other two races and you're there.

resol · 2009-03-04 04:48:05

That worked great!

Now can you go one better and arrange them so that no one (or hardly anyone) are in back to back races?

This is the best I can come up with. The ones with the asterisk are in back to back races.

On the remaining 2 races I just went with odd and even.

Race - 1 2 3 4 5 6 7 8
_____________________________
Cars - 1 *9 1 *5 1 *5 1 2
2 *10 2 *6 2 6 3 4
3 *11 3 *7 3 *7 *5 6
4 *12 4 *8 4 8 *7 8
*9 13 *5 9 *13 *13 9 10
*10 14 *6 10 *14 *14 11 12
*11 15 *7 11 *15 *15 *13 14
*12 16 *8 12 *16 *16 *15 16

I may be asking the impossible.
Thanks Again, Derek

mathsyperson · 2009-03-04 05:43:01

You certainly can't eliminate back-to-back racers entirely.
Let's say the first race involved cars 1-8. Then the next race would have to involve cars 9-16, and the race after that would have to involve 1-8 again. All the races after the first are forced, and so you'd have lots of cars that never faced each other.

I'd prioritise making sure that no car is in three consecutive races.

resol · 2009-03-04 09:24:23

Thanks again for your help. I think Im going to change a few and call it good.

Derek

Math Is Fun Forum

#1 2009-03-03 11:08:01

8 Races Between 16 Cars Question

#2 2009-03-03 11:27:36

Re: 8 Races Between 16 Cars Question

#3 2009-03-04 04:48:05

Re: 8 Races Between 16 Cars Question

#4 2009-03-04 05:43:01

Re: 8 Races Between 16 Cars Question

#5 2009-03-04 09:24:23

Re: 8 Races Between 16 Cars Question

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