In a certain tournament a player mist be defeated three times to be eliminated. If 512 contestants enter the tournament what is the greatest number of games that could be played?
Say you only have 3 players.
The maximum number is: 2+2+2+1+1
If 4 then: 2+2+2+2+1+1+1
If 5 then: 2+2+2+2+2+1+1+1+1
We see that general formula is, if n>=3 players then:
(2+2+...+2)+(1+1+...1)
Where 2 appears n times and 1 appears n-1 times.
Thus,
2n+1(n-1) = 3n - 1
Thus, if you have 512 that means,
3(512)-1
Assume that in a game one player wins and the other loses, that is no draws.
At the end of the tournament 511 players have each lost 3 games, and the
winner may have lost 0, 1 or 2 games. As each, and every game had a loser
then there were 3*511, 3*511+1 or 3*511+2 games, the largest of these is
3*511+2=3*512-1.
RonL