Chess Tournament

as the number of total players is 12 then the total number of match is 12C2=66. in each match the total count of points are 2. so the total number of points is 132.

The handshaking principle applies here very heavily. For each of the 12 players the player can play against 11 opponents. Since the matches are counted twice, dividing 12×11 by 2 will result in the number of matches. For each of these matches there are two possible outcomes: somebody wins and somebody loses, or the players tie. Both outcomes result in a point change of positive 2. So, the rule for any amount of players is the following:

f(p) = ((p(p-1))/2)2

Which simplifies to...

$f(p) = p^{2}-p$

Here's some Python , too.

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players = 12
points = 0
# for each match
for player in xrange(players):
    for opponent in xrange(players):
        if player < opponent:
            # somebody wins
            points += 3
            # somebody loses
            points -= 1
print "Points:", points

The number of points handed out per game is the same if there is a tie or a win. So assume that all games are tied, then think about how the league table will look at the end of the first round. All players will have played 11 games and will have 11 points. Therefore total points = 11 x 12 players = 132.

Let us define the state after each chess game by two variables - g (No of games played so far).and S (Sum of points of all players after g games)

The result after each game can be either having a - Case 1 : winner or looser where the winner contributes +3 to the sum and looser -1 so overall S becomes (S+2). Case 2 : tie where each player contributes +1 so S again becomes (S+2).

Taking S - 2 g = 0 as an invariant. The invariant holds in the base case when g = 0 and S = 0. Let us assume by induction the invariant holds after g games i.e. S - 2 g = 0 After g+1 game S becomes (S+2) so, (S+2) - 2 (g+1) = S + 2 - 2 g -2 = S - 2*g = 0. Hence proved that the invariant holds after g games.

No of games played in the first round = 12C2 = 66 games so S - 2 * (66) = 0 therefore S = 132

there are 66 games that are possible, and only 33 will be winners and 33 are lossers so 33x3 =99 plus the lossers = 33x1 = 33 summing up = 132

lets assume all matches played are tied, hence each player at the end of tournament gets 11 points, and hence all together points is(12x11)=132points