Games & Tournament type of questions often feature in CAT and could be of innumerable patterns. Below are some of the common ones which have been discussed with the help of an example. (As a reader feel free to request for any other type that you may want to understand, by mentioning it in the comment box below this post)
Round
Robin League
Whenever ‘n’ number of teams play once against each
other and the top ‘x’ proceed to the next round often the following questions
are asked –
i) What is the total number of matches played in the
round?
Ans. ^{n}C_{2}
ii) What is the minimum number of points that a team
should win in order to qualify for the next round (given that win = 2 pts; loss
= 0 pts; tie/draw = 1 pt split between both sides)?
Ans. Let (x1) team win maximum number of matches
possible and after deducting their points distribute the balance between the
rest of the teams, equally. As 1 team from the rest have to qualify for the
next round this process shall give us one team with minimum points qualifying
for the next round. (#1 first round)
iii) What is the maximum points with which a team
doesn’t qualify for the next round?
Ans. As any team qualifying for the next round
cannot get lesser points than a team failing to qualify, the maximum we can
give to a team not proceeding to the next round is equal to the teams going to
the next round. Here, attempt would be to give maximum points possible to (x+1)
teams equally. As only x teams shall move ahead to the next round, the team
which got equal points as top x teams shall have maximum points without being
able to qualify for the next round. (#1 first round)
KnockOut
Tournament
When ‘n’ number of teams play a tournament (‘n’
being an even number) with every team playing a match each, such that top
ranked team plays against the least ranked side and second ranked team play
against the second least ranked side and so on – the number of matches in that
round shall be n/2 and the sum of ranks of any two teams playing each other
shall be (n+1). (#1 second round)
Coin
Picking Game
When 2 players finish picking ‘n’ number of coins
turn by turn by picking minimum ‘x’ and maximum ‘y’ coins in their respective
turns then they choose a target number of coins to be left at end for the
opponent and maintain gap of (x+y) coins. (#1 final round)
#1.
32 top players of the world participate in a multigames tournament which would
test their intelligence, fitness and physical strength. The participants are
divided equally into 2 groups – A & B wherein everyone play once against
each player in their respective group. The winner of every game is awarded 2
points while the loser gets 0 points. In case of a draw, both players get 1
point each. At the end of all matches in both the groups, only the top 8 players
from each group proceed to the next round. In case, 2 or more players end up
with same number of points they are given priority based on their ranking given
to them prior to the tournament such that when two players have scored equal
points the player with a better ranking gets a higher position in the points
table.
A fresh ranking is given to all players reaching the
second round from 116 based on the points scored by them and their previous
rankings (this ranking shall be valid from second round, till the end of the
tournament). In the second round every player competes against exactly one player
in an arm wrestling duel wherein the player ranked 1^{st} (as per fresh
ranking after round 1) plays against 16^{th} rank in the first game, 2^{nd}
against 15^{th} in the second game and so on till 8^{th} plays
against 9^{th} rank. Only the winners proceed to the third round. Whenever
a player wins against another player ranked above it, then the lower ranked
player is said to have caused an upset.
In the third round, winners of 1^{st}, 3^{rd},
5^{th} and 7^{th} games of second round team up against another
team of winners of the rest of the games in the second round for a tug of war
contest. Only players of the winning side proceed to the next round.
In the fourth round all 4 players participate in a
swimming contest and the top 2 qualify for the final round.
In the final round 50 marbles are left on a table
and both players pick some marbles turn by turn with a minimum of 1 and maximum
of 5 marbles in each turn. The person who picks the last marble wins the tournament.
The first turn to pick the marbles would be decided by a toss, wherein the
winner of the toss may choose to take the first turn or ask the opponent to
pick first.
i)
What is the minimum points with which a player can qualify the second round?
Sol.
Total points from all matches in a group = ^{16}C_{2} x 2 = 240
Maximum points that can be scored by a player are 15
x 2 = 30.
Similarly maximum points that can be scored by top 7
players in a group are –
With 168 points out of 240 already assigned to top 7 teams, the balance can be distributed equally among the rest, giving the other 9 teams 8 points each. As some team has to take the 8^{th} position in the points table and qualify for the next round, the minimum points needed to do the same shall be 8.
15 wins

30 points

14 wins

28 points

13 wins

26 points

12 wins

24 points

11 wins

22 points

10 wins

20 points

9 wins

18 points

84 matches

168 points

With 168 points out of 240 already assigned to top 7 teams, the balance can be distributed equally among the rest, giving the other 9 teams 8 points each. As some team has to take the 8^{th} position in the points table and qualify for the next round, the minimum points needed to do the same shall be 8.
ii)
What is the maximum possible point scored by a team which does not qualify for
the second round?
Sol.
A team which doesn’t qualify must have equal or lesser points than each of the
teams which qualify. In order to maximize the score of a team not making it to
the second round we can maximize the points among top 9 teams (as only 8 shall
qualify to second round from a particular group).
Here we can begin by eliminating those points which
cannot be scored by the top 9 teams i.e. points from matches among the bottom 7
teams. The minimum points scored by bottom 7 teams together shall be ^{7}C_{2}
x 2 = 42 points.
The balance (240 – 42) = 198 points can be
distributed equally among the rest 9 teams by letting them win 198/9 = 22
points each.
As only 8 among the 9 top teams shall proceed to the
next round, the maximum points with which a team does not qualify for the
second round is 22.
iii)
If all even ranked players lose in the second round then what is maximum
difference between the ranks of any two players qualifying for the third round?
Sol.
If the even ranked players lose their respective games, then the list of
winners who qualify to the third round are –
Matches

Competing players

Winners


Match 1

1

16

1

Match 2

2

15

15

Match 3

3

14

3

Match 4

4

13

13

Match 5

5

12

5

Match 6

6

11

11

Match 7

7

10

7

Match 8

8

9

9

The maximum difference the ranks of any two players
qualifying to the third round is
(15 – 1) = 14
iv)
What is the lowest possible rank of a player who won the tournament by beating
the 1^{st} ranked player in the final round?
Sol.
As player ranked 1 reached the final round, it is clear that winners of 1^{st},
3^{rd}, 5^{th} and 7^{th} matches must have won the
third round.
The lowest rank available in the second round was
16, but as it was scheduled to play against 1 (which reached the final as per
the question) it could not have reached the third round.
Player ranked 15 would have played the second match
of the second round and even if it won the second round match it would feature
in the opponent team of 1 in the third round of tug of war. Hence it could not
have reached the final.
Player ranked 14 would have played the third match
of the second round and could have reached the final and won against 1. Hence
the answer is 14.
v)
If the player ranked 14 won the toss in the final round and picked 1 coin in
the first turn and did not make any mistake after that, will (s)he cause an
upset?
Sol.
As 14 could also be playing the final round against 16 (a player ranked below
itself), even if (s)he goes on to win, it may or may not be an upset. Hence
answer to this question can’t be determined.
vi)
If the best possible ranked players reached the final round (such that sum of
their ranks is lowest possible among the combination of players that could
reach the final round together), and the player ranked lower among them won the
toss then what must be her/his decision to ensure her/his victory? (it is to be
noted that both players have to pick the number of marbles equal to their rank
in their respective first turns in the final round)
Sol.
The lowest possible sum total of ranks of two players who can reach finals
together shall be 4, when player ranked 1 and 3 reach the final round.
As the number of marbles to be picked in respective
first turns have already been defined, we know that after both players would have
picked once, they would remove (1 + 3) = 4 marbles together and leave 46 on the
table for the player who picked first to take the next turn.
In order to pick the last marble a player would have
to leave exactly 6 marbles on the table for her/his opponent at the end to ensure
victory. From there no matter what number of marbles the opponent picks one can
ensure its win in the last turn (refer below table):
6 marbles are left on the table


Opponent picks

Winner picks

5

1

4

2

3

3

2

4

1

5

This implies that from any position, a player can
maintain a sum of 6 marbles (which is sum of minimum and maximum limit of total
marbles to be picked in a turn i.e. 5 + 1) for the opponents turn followed by
her/his turn.
In this case as player ranked 3 shall win the toss,
(s)he can definitely ensure her/his victory only by leaving 6 or any multiple
of 6 marbles for the opponent. Hence player ranked 3 shall choose to pick first,
and after both players have picked the stipulated number of marbles (i.e. 3+1 =
4) in respective first turns, player ranked 3 shall pick 4 marbles to leave the
opponent with 42 marbles (a multiple of 6) and ensure her/his win from there
on.