CAT Question Bank  Games & Tournaments

Q18) A game is played on a square grid made of a hundred squares the bottom left hand corner square being numbered 1 and the top left hand corner square being numbered 100. The numbering is done sequentially, from left to right in the bottom row and from right to left in the row above and so on. The game ends when a counter reaches or crosses the square numbered 100. The game is played with a pair of dice (six faced). If one of the dice shows two and the other shows six, then the player has to move forward by 6 + 2 = 8 squares. If a player is on square number 96, what is the probability that the game ends after one throw of the dice?

Q19) Seven Hockey teams A, B, C, D, E, F and G participated in a tournament. Each team played with every other team twice and D won all of its games and F lost all of its games. B and G scored equal points and are ahead of C. C, A and E also scored equal number of points. Each team gets 3 points, 1 point and no point for a win, draw and loss respectively. What is the highest possible number of points that can be scored by team A?
a.14
b. 15
c. 16
d. 17[OA: 17]

Total number of matches = 42
Total points without draw = 42 * 3 = 126
with draw point reduces by 1
because winlose  point distribution = 3
draw draw= point distribution=2
Let the total points of A,C,E be a and that of B and G be b
3a + 2b + 36 = Total points [ also a < b ]
Total points can be 126 or 125 or 124 and so on
But our objective is to maximize a we have to take value of total as high as possible.
3a + 2b = 126 then max = 16
but if we take 3a + 2b = 125 then max(a) = 17
so 17[Credits : @hemant_malhotra]

Q20) From a group of N players, coach has to select a captain. Even after holding a series of meetings, the team management and the players failed to reach a consensus. It was then proposed that all N players be given a number from 1 to N. Then they will be asked to stand on a podium in a circular arrangement, and counting would start from the player numbered 1. The counting would be done in a clockwise fashion. The rule is that every alternate player would be asked to step down as the counting continued, with the circle getting smaller and smaller, till only one person remains standing. Therefore the first person to be eliminated would be the player number 2. If N is 545, which position should a player choose if he has to be the captain?
(a) 3
(b) 67
(c) 195
(d) 323
(e) 451[OA: 67]

Q21) A dartboard is divided into 4 concentric circles having radii of 1 cm, 2 cm, 3 cm and 4 cm respectively. The prize amounts for hitting the circles (starting from the innermost circle)are Rs. 40, Rs. 30, Rs. 20 and Rs. 10 respectively. Every participant has to purchase a ticket in order to be able to throw the dart once. How much amount should the organizer of the game charge per ticket so as to ensure that he will end up with noprofit and noloss situation? (Assume that a large number of participants participate in the game and every participant in the game randomly throws the dart towards the dartboard and always hits the dartboard.)
a) Rs. 23.25
b) Rs. 27.00
c) Rs. 18.75
d) Rs. 16.50[OA: 18.75]

Q22) Crap is a popular game in which you throw a pair of dice one or more times until you either win or lose. There are two ways to win in the game. You can throw the dice once and obtain a score of 7 or 11 in the first throw, or you can obtain a 4,5,6,8,9, or 10 on the first throw and repeat the same score on the subsequent throw before you obtain a 7. There are two ways to lose. you can throw the dice once and obtain a 2,3, or 12, or you can obtain a 4, 5, 6, 8, 9, or 10 on the first throw and then obtain a 7 on a subsequent throw before you repeated your original score.
What is the probability that a person wins on the 2nd throw?
a) 100/1296
b) 50/1296
c) 100/216
d) 50/216
e) None of theseWhat is the probability that a person loses on the 2nd throw ?
a) 100/1296
b) 120/1296
c) 144/1296
d) 72/1296
e) None of these

Q23) Four friends A, B, C and D are playing a game “Pass it on”. Initially each of them has 24 coins. The game starts with A passing a coin to B, then B passing 2 coins to C, then C passing 3 coins to D and then D passing 4 coins to A . At this point, one round is completed. In the 2nd round, A passes 5 coins to B, B passes 6 coins to C and so on. The game ends when one person has all the coins and he is declared the winner. Find the number of coins with B in the 8th round just after he has received the coins from A.
a) 32
b) 44
c) 45
d) 46

Q24) There are 111 players participating in a singles tennis tournament. The player who is losing will be out of the tournament. For each and every match, One new ball is taken. Find the no. of balls required for the entire tournament ?

Q25) In a mathematical game, one hundred people are standing in a line and they are required to count off in fives as one, two, three, four, five, one two, three, four, five and so on from the first person in the line. the person who says five is taken out of the line. those remaining repeat this procedure until only four people are left in the line. What was the original position in the line of the last person to leave?
a. 93
b. 96
c. 97
d. 98[OA: D]

Q26) In a chess tournament, every player plays one match against every other player. Each match can result in a draw or win to one player anddefeat of the other player. Players are awarded 1 point for every win, 0 points for every defeat and 0.5 points for every draw. In all, 4Russians and more than one American were the only participants in the tournament. Russian players together scored 21 points in the tournament and all American players scored equal number of nonzero integer number of points. How many points were scored by each American player?
a) 10
b) 8
c) 3
d) More than one answer possible

Q27) There are 16 teams and they are divided into 2 pools of 8 each. Each team in a group plays against one another on a roundrobin basis. Draws in the competition are not allowed. The top four teams from each group will qualify for the next round i.e round 2. In case of teams having the same number of wins, the team with better runrate would be ranked ahead.
 Minimum number of wins required to qualify for the next round ?
 Minimum number of wins required to guarantee qualification in the next round ?


1 group is consisting of 8 teams. So each team will play 7 match each. Suppose each of the 8 teams were seeded and we consider the case where a higher seeded team will always win.
So the number of wins for the 8 teams would be 7,6,5,4,3,2,1,0 with highest seeded team winning all and lowest seeded team losing all.
For minimum number of wins we allow 3 teams to win maximum number of matches. Of the remaining 5 teams just find out the mean of their number of wins.
In this case it would be (4+3+2+1+0)/5=2.
So 5 teams can end up with 2 wins each and a team with better run rate will qualify with 2 wins. 
In this case consider the mean of first 5 higher seeded teams
(7+6+5+4+3)/5=5
So it may be the case that 5 teams can end up having 5 wins each. And hence 1 team will miss the second round birth. So minimum number of wins to guarantee a place would be 6.


Q28) Two teams P and Q contest in a game of tug of war. Team P has ‘n’ players with an average weight of ‘a’ kgs, ‘a’ being an integer. Team Q also has ‘n’ players with an average weight of ‘a’ kgs. The rule of the game is that the team that wins two rounds consecutively wins the game. However, once a team wins a round, the other team adds one more player to their team such that their average weight increases by 1 kg for the next round. The game progressed as follows.
Round 1: Team Q won
Round 2: Team P won
Round 3: Team Q won
Round 4: Team P won
Round 5: Team Q won
Round 6: Team P won
Round 7: Team P won
Which of the following can be the sum of the weights of the new players added in team P?
(a) 248
(b) 250
(c) 249
(d) 247[OA: C]

First player to be added brings in an excess of 1 kg for each player (including himself) over the average, a.
This means his weight is a+n+1.
The second player to be added brings in an excess of 1 for all including himself over the previous average, a+1. This means his weight is a+1+n+2
Third player weighs a+2+n+3.
Total weight is 3a + 3n + 9. This is divisible by 3. So the total can be 249.
[@rajesh_balasubramanian ]

Q29) Ram and Shyam were playing a game that involved picking up marbles from a table. Each player in his turn has to pick up at least 2 or at most 7 marbles, except when there is only one marble left on the table in which case, he has to pick up that marble. The game ends when all the marbles on the table are picked up. If the game starts with 62 marbles and it is Ram's turn to play, then how many marbles should he pick up so as to ensure his win?
Assume that both play intelligently so as to win the game.
Assume that the player who picks the last marble LOSES the game.
a. 4 or 3
b. 6 or 7
c. 5 or 4
d. 2 or 1

Q30) In a soccer tournament n teams play against one another exactly once. The win fetches 3 points, draw 1 each and loss 0. After all the matches were played, it was noticed that the top team had unique number of maximum points and unique least number of wins. What can be the minimum possible value of n?
a) 5
b) 6
c) 7
d) 8
e) none of these

@rowdyrathore Shouldnt it be smallest value of n greater than 5 , or any other limiting factor?

@rowdyrathore We select two points out of 98 to make a chord, given by 98C2. This gives 4753 , which leaves a remainder on dividing by 3 .thus anthony wins.

@rowdyrathore The rth match in Round 1 will be played between Seed ‘r’ and Seed ‘n+1r’
so 7th match will be played between seed 7 and seed 32+17=26
and 10 th match will be played between seed 10 and seed 32+110=23

@rowdyrathore please provide solution