CAT Question Bank - Games & Tournaments
Q5) In a game played by two people there were initially N match sticks kept on the table. A move in the game consists of a player removing either one or two matchsticks from the table. The one who takes the last matchstick loses. Players make moves alternately. The player who will make the first move is A. The other player is B. The largest value of N (greater than 5) that ensures a win for B is?
Q6) There are 98 given points on a circle. Amar, Akbar and Anthony start playing a game by drawing a chord one by one between two of the points which have not yet been joined together. The game ends when all such points have been joined exhaustively. The winner is the one who draws the last chord. If Anthony starts the game, followed by Akbar, and then Amar, then who will win?
Q7) In a game, the winner awarded "W" points and the loser "L" points, where W and L are natural numbers such that W > L > 0. Bunty and Babli played a series of such games and it was found that the sum of their scores points is 60 in the end. Babli won exactly 2 games. If there was no tie in any game and Bunty had more points than Babli in the end, then find the maximum value of W.
Q8) Four people A, B, C and D are playing a game. Each of them starts with a different number of points. At the end of each round, one Winner is declared. Each of the other three players gives exactly half his points to the Winner. After three rounds. the scores for A, B, C and D respectively are 42, 12, 4 and 24.
- Who won the first round? ‘
- Who won the second round?
- How many points did B have at the start of the game?
Q9) Raju and Radhika are playing a game which involves picking up coins kept on a table. Each person in his/ her turn has to pick a minimum of 1 and a maximum of 7 coins, until all the coins are picked up by the players. Assume that both players are playing intelligently with the intention of winning the game. If there are 42 coins on the table and it is Raju’s turn to play, then how many coins should he pick up to ensure his win?
Assume that the player who picks the last coin win the game.
Q10) Abhishek and Ajit are two friends playing a game where they throw a fair die alternately. The player who first throws a six wins the game. The total prize money is Rs. 110 which is split among the two friends in the ratio of number of games won. If Abhishek has the first throw, how much money does Ajit expect to win (in Rs.)?
Q11) A game involving a biased die is such that ₹ 5 is paid each time the die shows up a score of 3, while ₹ 8 is paid for every other score on the die. The die is such that the score of 3 occurs 4 times as frequently as any other score. How much would a person be willing to pay as entry fee each time , if in the long run , there has to be neither profit nor loss for taking part in this game?
Q12) A gaming company has created a computer based cricket game, in which a player can face different types of balls and hit different shots on them. Depending on the choice of shot and the timing, a player can score 2, 3, 4 or 6 runs. If the player misses the shot, a wicket is lost, which also causes a deduction of 5 runs in the score. No other result is possible on a ball. If a player faces 60 balls in a game, which of the following is not a possible score at the end of the game?
A) 280 runs with 7 wickets lost
B) 290 runs with 6 wickets lost
C) 271 runs with 8 wickets lost
D) 259 runs with 9 wickets lost
The maximum score possible in the game is 360 (60 balls × 6 runs) with no wickets lost.
For each wicket lost, this score decreases by 11 (6 runs not scored plus 5 runs lost due to the wicket). For example, if 3 wickets are lost, the maximum possible score is 327 [360 − (3 × 11)], which could be achieved by hitting sixes on all the remaining 57 balls.
For 7, 6, 9 and 8 wickets lost (as given in the options), the maximum possible scores are 283, 294, 261 and 272 respectively.
The scores in options 1 and 2 can be achieved if a player hits 3 or 2 runs instead of 6 on one ball. However, option 3 is not possible, as for that, a player would have to hit 5 runs instead of 6 on one ball, which is not possible in this game.
Hence, option C.
[Credits : Lokesh Agarwal]
Q13) A game is played between 2 players and one player is declared as winner. All the winners from first round, played in the second round. All the winners from second round played in third round and so on. If 8 rounds were played to declare only one player as winner, how many players played in first round?
Q14) Six men engaged in a game. Whenever a player won a game he doubled the money of each of the players. That is, he gave each player as much money as he/she originally had in his/her pocket. They played six games and each player won one game. In the end all of them had Rs. 256 each. What was the difference between the money the person with the highest amount had and the person with the smallest amount had to start off with?
Q15) In the game of Dubblefud, red chips, blue chips and green chips are each worth 2,4 and 5 points respectively. In a certain selection of chips, the product of the point values of the chips is 16,000. If the number of blue chips in this selection equals the number of green chips, how many red chips are in the selection?
case 1: we are selecting no couple.
2 males can be selected in 4c2 = 6 ways.
We cannot select the wives of already selected males and should go for others.
so 2c2=1 way.
So in total 2 males and 2 females can be selected in 4c2 * 2c2 = 6 ways..
now the teams can be interchanged as well.
so 6 * 2 = 12 games possible.
case 2: when exactly one couple is selected.
so u can select that couple in 4c1=4 ways.
Now after this u need to select 1 male and 1 female more but they should not be a couple.
so selecting 1 male from remaining 3 males in 3c1 way and now for a female u have 2 options left..
so 4c1 * 3c1 * 2 = 24 ways
case 3: when the 2 couples are selected. so 4c2 = 6 ways
Total = 42
Note : Assume A1 B1 and A2 B2 are the two couples selected. Now they cannot be in same team but can play in the same game. A1B2 in one team and A2B1 in other team
[Credits: Jasneet Dua]
Q16) Ashwin and Vijay are playing a game wherein each of them have to choose any no. between 2 and 5. For e.g. if Ashwin picks up 4 then Vijay can next pick from 6 and 9; following which Ashwin can again choose from 8 and 14 depending on what the former person has chosen. The person who chooses 60 first wins.
Given that Ashwin starts the game what no should he choose first ?
(d) he can never win
Ashwin starts first and chooses 10; what should Vijay choose next in order to win?
(e) None of the above
Q17) Four players – P, Q, R and S – played a game of bursting balloons by shooting at them. In this game the following terms are defined.
Aim = One attempt of shooting at a balloon.
Shot = One instance of shooting down a balloon.
Miss = One wasted aim.
The rules of the game are as follows:
(i) Each person will be given a maximum of four rounds of aims, the first round comprising three aims. A person gets the second round of aims only if he scores at least one shot in the first (i.e. the previous) round and so on.
(ii) ( If in any round, the number of shots by a player is 50% or more but less than 100% of the number of aims he had in that round, then he gets one extra aim in each of the remaining rounds. If the number of shots by the player is 100% of the number of aims he had in that round, he gets two extra aims in each of the remaining rounds.)
(iii) For each shot, a player is awarded five points and for each miss, he earns two negative points.
If the number of shots in P’s first round = that in Q’s third round = that in S’s second round = that in R’s fourth round, and P scored eight points in the second round, then what is the maximum possible score by Q in the fourth round?
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?
Total number of matches = 42
Total points without draw = 42 * 3 = 126
with draw point reduces by 1
because win-lose - 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
[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?
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 no-profit and no-loss 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