Maths Puzzle
Q.No: 1
Test Name : CAT Paper 2003 (R)
Directions for questions 51 to 53: Answer the questions on the basis of the information given below. The seven basic symbols in a certain numeral system and their respective values are as follows:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then the smaller value is subtracted from the larger.
For example, XLVI = (50 10) + 5 + 1 = 46.
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then the smaller value is subtracted from the larger.
For example, XLVI = (50 10) + 5 + 1 = 46.
The value of the numeral MDCCLXXXVII is
Solution:
MDCCLXXXVII = 1000 + 500 + 100 + 100 + 50 + 10 + 10
+ 10 + 5 + 1 + 1 = 1787
Q.No: 2
Test Name : CAT Paper 2003 (R)
Directions for questions 51 to 53: Answer the questions on the basis of the information given below. The seven basic symbols in a certain numeral system and their respective values are as follows:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then the smaller value is subtracted from the larger.
For example, XLVI = (50 10) + 5 + 1 = 46.
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then the smaller value is subtracted from the larger.
For example, XLVI = (50 10) + 5 + 1 = 46.
The value of the numeral MCMXCIX is
Solution:
MCMXCIX = 1000 + (1000 100) + (100 10) +
(10 1) = 1000 + 900 + 90 + 9 = 1999
Q.No: 3
Test Name : CAT Paper 2003 (R)
Directions for questions 51 to 53: Answer the questions on the basis of the information given below. The seven basic symbols in a certain numeral system and their respective values are as follows:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then the smaller value is subtracted from the larger.
For example, XLVI = (50 10) + 5 + 1 = 46.
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000
In general, the symbols in the numeral system are read from left to right, starting with the symbol representing the largest value; the same symbol cannot occur continuously more than three times; the value of the numeral is the sum of the values of the symbols. For example, XXVII = 10 + 10 + 5 + 1 + 1 = 27. An exception to the left-to-right reading occurs when a symbol is followed immediately by a symbol of greater value; then the smaller value is subtracted from the larger.
For example, XLVI = (50 10) + 5 + 1 = 46.
Which of the following represent the numeral for 1995?
I. MCMLXXV II. MCMXCV III. MVD IV. MVM
Solution:
(I) MCMLXXV = 1000 + (1000 100) + 50 + 10 + 10 + 5 = 1975
(II) MCMXCV = 1000 + (1000 100) + (100 10) + 5 = 1995(III) MVD = 1000 + (500 5) = 1495(IV) MVM = 1000 + (1000 5) = 1995Therefore, the answer is (II) and (IV), i.e. option (3).
Q.No: 4
Test Name : CAT Paper 2003 (R)
Directions for questions 94 to 96: Answer the questions on the basis of the tables given below.
Two binary operations ⊕ and * are defined over the set {a, e, f, g, h} as per the following tables:
Thus, according to the first table f ⊕ g = a , while according to the second table g * h = f , and so on. Also, let f2 = f * f, g3 = g * g * g , and so on.
Two binary operations ⊕ and * are defined over the set {a, e, f, g, h} as per the following tables:
Thus, according to the first table f ⊕ g = a , while according to the second table g * h = f , and so on. Also, let f2 = f * f, g3 = g * g * g , and so on.
What is the smallest positive integer n such that gn = e?
Solution:
Q.No: 5
Test Name : CAT Paper 2003 (R)
Directions for questions 94 to 96: Answer the questions on the basis of the tables given below.
Two binary operations ⊕ and * are defined over the set {a, e, f, g, h} as per the following tables:
Thus, according to the first table f ⊕ g = a , while according to the second table g * h = f , and so on. Also, let f2 = f * f, g3 = g * g * g , and so on.
Two binary operations ⊕ and * are defined over the set {a, e, f, g, h} as per the following tables:
Thus, according to the first table f ⊕ g = a , while according to the second table g * h = f , and so on. Also, let f2 = f * f, g3 = g * g * g , and so on.
Upon simplification, f ⊕[f * {f ⊕(f * f )}] equals
Solution:
Q.No: 6
Test Name : CAT Paper 2003 (R)
Directions for questions 94 to 96: Answer the questions on the basis of the tables given below.
Two binary operations ⊕ and * are defined over the set {a, e, f, g, h} as per the following tables:
Thus, according to the first table f ⊕ g = a , while according to the second table g * h = f , and so on. Also, let f2 = f * f, g3 = g * g * g , and so on.
Two binary operations ⊕ and * are defined over the set {a, e, f, g, h} as per the following tables:
Thus, according to the first table f ⊕ g = a , while according to the second table g * h = f , and so on. Also, let f2 = f * f, g3 = g * g * g , and so on.
Upon simplification, {a10 * (f10 ⊕ g9 )} ⊕ e8 equals
Solution:
Q.No: 7
Test Name : CAT Paper 1990
Consider the following steps :
1. Put x = 1, y = 2
2. Replace x by xy
3. Replace y by y +1
4. If y = 5 then go to step 6 otherwise go to step 5.
5. Go to step 2
6. Stop
Then the final value of x equals
Solution:
Q.No: 8
Test Name : CAT Paper 1990
116 people participated in a singles tennis tournament of knock out format. The players are paired up in the first round, the winners of the first round are paired up in second round, and so on till the final is played between two players. If after any round, there is odd number of players, one player is given a bye, i.e. he skips that round and plays the next round with the winners. Find the total number of matches played in the tournament.
Solution:
There are 116 players in all. If we have to choose 1
winner, there have to be 115 losers in all. And since 1
match gives 1 loser, there has to be 115 matches to be
played in all in the tournament.
Q.No: 9
Test Name : CAT Paper 1990
Questions 82-84 : The following questions relate to a game to be played by you and your friend. The game consists of a 4 x 4 board (see below) where each cell contains a positive integer. You and your friend make moves alternately. A move by any of the players consists of splitting the current board configuration into two equal halves and retaining one of them. In your moves you are allowed to split the board only vertically and to decide to retain either the left or the right half. Your friend, in his/her moves, can split the board only horizontally and can retain either the lower or the upper half. After two moves by each player a single cell will remain which can no longer be split and the number in that cell will be treated as the gain (in rupees) of the person who has started the game. A sample game is shown below.
So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
If you choose (retain right) (retain left) in your turns, the best move sequence for your friend to reduce your gain to a minimum will be
Solution:
Q.No: 10
Test Name : CAT Paper 1990
Questions 82-84 : The following questions relate to a game to be played by you and your friend. The game consists of a 4 x 4 board (see below) where each cell contains a positive integer. You and your friend make moves alternately. A move by any of the players consists of splitting the current board configuration into two equal halves and retaining one of them. In your moves you are allowed to split the board only vertically and to decide to retain either the left or the right half. Your friend, in his/her moves, can split the board only horizontally and can retain either the lower or the upper half. After two moves by each player a single cell will remain which can no longer be split and the number in that cell will be treated as the gain (in rupees) of the person who has started the game. A sample game is shown below.
So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
If both of you select your moves intelligently then at the end of the game your gain will be
Solution:
Q.No: 11
Test Name : CAT Paper 1990
Questions 82-84 : The following questions relate to a game to be played by you and your friend. The game consists of a 4 x 4 board (see below) where each cell contains a positive integer. You and your friend make moves alternately. A move by any of the players consists of splitting the current board configuration into two equal halves and retaining one of them. In your moves you are allowed to split the board only vertically and to decide to retain either the left or the right half. Your friend, in his/her moves, can split the board only horizontally and can retain either the lower or the upper half. After two moves by each player a single cell will remain which can no longer be split and the number in that cell will be treated as the gain (in rupees) of the person who has started the game. A sample game is shown below.
So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
So your gain is Re.1. With the same initial board configuration as above and assuming that you have to make the first move, answer the following questions.
If your first move is (retain right), then whatever moves your friend may select you can always force a gain of no less than
Solution:
If your first move is (retain right) then the grid will look the same as in Q82. Your friend may hence choose either (retain
upper), which will leave you to choose from 2,4,6,7 or he may choose (retain lower), which will leave you to choose
from 3,2,8,4. In case he takes the former move, you can then move (retain right) and hence force a minimum gain of 4.
But in case he chooses the latter move, you can then move (retain move) and force a minimum gain of 3. In either case
you can force a minimum gain of Rs.3
Q.No: 12
Test Name : CAT Paper 1990
Questions 91-94 : The pages of a book are numbered 0, 1, 2 … upto M, M>0. There are four categories of instructions that direct a person in positioning the book at a page. The instruction types and their
meanings are :
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
FORWARD, 25 ; BACKWARD, 10.
Which of the following statements is true about the above set of instructions?
Solution:
Q.No: 13
Test Name : CAT Paper 1990
Questions 91-94 : The pages of a book are numbered 0, 1, 2 … upto M, M>0. There are four categories of instructions that direct a person in positioning the book at a page. The instruction types and their
meanings are :
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
BACKWARD, 5; FORWARD, 5. Which of the following statements is true about the above set of instructions?
Solution:
Q.No: 14
Test Name : CAT Paper 1990
Questions 91-94 : The pages of a book are numbered 0, 1, 2 … upto M, M>0. There are four categories of instructions that direct a person in positioning the book at a page. The instruction types and their
meanings are :
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
FORWARD, 10; FORWARD, 10. Which of the following statements about the above instructions is true?
Solution:
Q.No: 15
Test Name : CAT Paper 1990
Questions 91-94 : The pages of a book are numbered 0, 1, 2 … upto M, M>0. There are four categories of instructions that direct a person in positioning the book at a page. The instruction types and their
meanings are :
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
1. OPEN : Position the book at page No. 1
2. CLOSE : Position the book at page No. 0
3. FORWARD, n :From the current page move forward by n pages; if, in this process, page number M is reached, stop at M.
4. BACKWARD, n : From the current page, move backward by n pages; if in this process, page number 0 is reached, stop at page number 0.
In each of the following questions, you will find a sequence of instructions formed from the above categories. In each case, let n1 be the page number before the instructions are executed and n2 be the page number at which the book is positioned after the instructions are executed.
FORWARD, 5; BACKWARD, 4. Which of the following statements about the above instructions is true?
Solution:
Solution:
MDCCLXXXVII = 1000 + 500 + 100 + 100 + 50 + 10 + 10
+ 10 + 5 + 1 + 1 = 1787
Solution:
MCMXCIX = 1000 + (1000 100) + (100 10) +
(10 1) = 1000 + 900 + 90 + 9 = 1999
Solution:
(I) MCMLXXV = 1000 + (1000 100) + 50 + 10 + 10 + 5 = 1975
(II) MCMXCV = 1000 + (1000 100) + (100 10) + 5 = 1995(III) MVD = 1000 + (500 5) = 1495(IV) MVM = 1000 + (1000 5) = 1995Therefore, the answer is (II) and (IV), i.e. option (3).
Solution:
Solution:
Solution:
Solution:
Solution:
There are 116 players in all. If we have to choose 1
winner, there have to be 115 losers in all. And since 1
match gives 1 loser, there has to be 115 matches to be
played in all in the tournament.
Solution:
Solution:
Solution:
If your first move is (retain right) then the grid will look the same as in Q82. Your friend may hence choose either (retain
upper), which will leave you to choose from 2,4,6,7 or he may choose (retain lower), which will leave you to choose
from 3,2,8,4. In case he takes the former move, you can then move (retain right) and hence force a minimum gain of 4.
But in case he chooses the latter move, you can then move (retain move) and force a minimum gain of 3. In either case
you can force a minimum gain of Rs.3
Solution:
Solution:
Solution:
Solution:
