Math Is Fun Forum

mrpace

Member

Registered: 2012-08-16

Posts: 88

How many matches will be played in this tournament?

Hi there

18 players are playing in a tennis tournament (1 vs 1)

There are two criteria

1) Each player must play exactly 8 matches
2) A player can not play against the same player more than once

I've only ever seen problems where everyone plays everyone which are always straight forward.

Not sure how to approach this one.

Thanks

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: How many matches will be played in this tournament?

hi mrpace,

What are we actually after here. Do you want a list of 72 matches (That's [18 x 8]/2 ) ?

If every player plays everyone else just once that's (18 x 17)/2 = 153 matches. So there's lots of pairs that never occur.  There are loads of ways to come up with a list; I'd probably write a program to do it as it's somewhat tedious.

If in doubt choose a simpler example first.

Let's say there are just six players  and each player must play just 3 matches.

AB, AD, AF, BC, BE, CD, CF, ED, EF  is one solution but there are many others. To get this without lots of trial and error, I made a six by six table with A -F across the top and down the side. I put a cross in the leading diagonal as a player cannot play themselves.  Then I chose opponents for A, mixed it about a bit for B and so on until I had 3 ticks in each row and column.  By, for example, switching the As and Ds, you get another, different solution, which is why I say there are lots.

I'm not sure what else to say here.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 243

Re: How many matches will be played in this tournament?

hi mrpace,

I draw an 18x18 table on Excel. I filled its diagonal with 18 zeros.
I let its below part be a replica of its upper one, since (P1,P2) = (P2,P1) for example.

By filling the rows of upper cells with 8 one's for each player, I got the total of matches equal to 144 (8*18).
And to check that every player plays 8 matches, I summed the cells of each row/player (horizontally) and of each column/player (vertically). It was possible to let every sum be 8.

Kerim

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Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.