Each digit, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter A, B, C, D, E, F, G, H and I but not necessarily in this order. Further, each of A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
Determine the value of each of the alphabets such that the above conditions are satisfied.

A survey is conducted in a school among 150 students and it was found that 100 students play cricket, 90 students play football and 80 students play hockey. As physical education was given emphasis in the school, every student had to play at least one sport.
What can be the maximum number of students who play exactly two sports?
What can be the maximum number of students who play exactly one sport?

Prove or Disprove:

There exists some power of 3 which ends in 001.

How many 5 digit numbers are there such that the sum of its digits is odd?
Example: 12345 is a five digit number such that the sum of its digits = 1 + 2 + 3 + 4 + 5 = 15 = an odd number.

Prove that:
"In any group of 6 people, there are either 3 mutual friends or 3 mutual strangers".

Mr. A, Mr. B, Mr. C and Mr. D are four men who are Physicist, Mathematician, Doctor and Policeman, though not necessarily in that order. Mr. A and Mr. B are neighbours and they go together to their work by a bike. Mr. B makes more money than Mr. C. Mr. A meets Mr. D regularly in chess playing. The Physicist always goes by car. The only time the Doctor met the Policeman ever was when the Policeman arrested the Doctor for speeding. The policeman makes more money than the Mathematician. The Doctor makes more money than Mr. D

Who is the Physicist?
(a) Mr. A
(b) Mr. C
(c) Mr. D
(d) Mr. B
(e) Cannot be determined

Who is the second highest earner?
(a) Mr. C
(b) Mr. D
(c) Mr. A
(d) Either (a) or (b)
(e) Cannot be determined

Who is the Mathematician?
(a) Mr. A
(b) Mr. D
(c) Mr. B
(d) Mr. C
(e) Cannot be determined

What is the profession of Mr. B?
(a) Physicist
(b) Policeman
(c) Mathematician
(d) Doctor
(e) Cannot be determined

Let us suppose that there are 100 students in my class. Prove that among them there are at least two of them who have an equal number of friends.
Note: If A is friends with B then it also implies that B is friends with A or else A and B both are strangers to each other.

In a street there are 5 houses painted 5 different colours. In each house lives a person of a different nationality. The 5 homeowners each drink a different beverage, smoke a different brand of cigar, and keep a different pet.
1. The Brit lives in the Red house.
2. The Swede has a Dog.
3. The Dane drinks Tea.
4. The Green house is on the left of the White house.
5. The owner of the Green house drinks Coffee.
6. The person who smokes Pall Mall has Birds.
7. The owner of the Yellow house smokes Dunhill.
8. The man living in the center house drinks Milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blends lives next to the one who has Cats.
11. The man who has Horses lives next to the man who smokes Dunhill.
12. The man who smokes Blue Master drinks Beer.
13. The German smokes Prince.
14. The Norwegian lives next to the Blue house.
15. The man who smokes Blends has a neighbour who drinks water.

Find out the correct combination of drinks, pets, smokes and colour of the house for each of the five persons.

A very long hallway has 1000 doors numbered 1 to 1000; all doors are initially closed. One by one, 1000 people get into the hall; the first person opens each door, the second closes all door with even numbers, the third person closes door 3, opens door 6, closes door 9, opens door 12 etc. That is, the nth person changes all doors whose numbers are divisible by n. After all 1000 people have got into the hall, find out how many doors will remain open.

In the shooting of the movie, "A Beautiful Mind", Ravi, the actor, is running towards a vertical mirror with a speed of 10 m/sec on a line which is perpendicular to the plane of the mirror. The mirror is moving towards Ravi with a speed of 6 m/sec along the same line. Find the speed of Ravi’s image with respect to Ravi (the image of Ravi forms on the other side of the mirror such that the distance between the image of Ravi and the mirror is equal to the distance between Ravi and the mirror).

There are 60 workers who work for a company, out of which 25 are women. Also:
28 workers are married
26 workers are graduate
20 married workers are graduate of which 9 are men
15 men are graduate
15 men are married
How many unmarried women are graduates and how many unmarried women work in the company?

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

At what time do Ram and Shyam first meet each other?
(a) 10 a.m.
(b) 10:10 a.m.
(c) 10:20 a.m.
(d) 10:30 a.m.

At what time does Shyam overtake Ram?
(a) 10:20 a.m.
(b) 10:30 a.m.
(c) 10:40 a.m.
(d) 10:50 a.m.

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of the white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :
(a) 10
(b) 12
(c) 14
(d) 16

Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number is odd, and the number 9 appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he has to make before he can be certain to succeed?
a. 10,000
b. 2,430
c. 3,402
d. 3,006

Help Distress (HD) is an NGO involved in providing assistance to people suffering from natural disasters. Currently, it has 37 volunteers. They are involved in three projects: Tsunami Relief (TR) in Tamil Nadu, Flood Relief (FR) in Maharashtra, and Earthquake Relief (ER) in Gujarat. Each volunteer working with Help Distress has to be involved in at least one relief work project.
• A Maximum number of volunteers are involved in the FR project. Among them, the number of volunteers involved in FR
project alone. is equal to the volunteers having additional involvement in the ER project.
• The number of volunteers involved in the ER project alone is double the number of volunteers involved in all the three projects.
• 17 volunteers are involved in the TR project. ,
• The number of volunteers involved in the TR project alone is one less than the number of volunteers involved in ER project
alone.
• Ten volunteers involved in the TR project are also involved in at least one more project.

Based on the information given above, the minimum number of volunteers involved in both FR and TR projects, but not in the ER project is:
(a) 1
(b) 3
(c) 4
(d) 5

Which of the following additional information would enable to find the exact number of volunteers involved in various projects?
(a) Twenty volunteers are involved in FR.
(b) Four volunteers are involved in all the three projects.
(c) Twenty three volunteers are involved in exactly one project.
(d) No need for any additional information.

After some time, the volunteers who were involved in all the three projects were asked to withdraw from one project. As a result, one of the volunteers opted out of the TR project, and one opted out of the ER project, while the remaining ones involved in all the three projects opted out of the FR project. Which of the following statements, then, necessarily follows?
(a) The lowest number of volunteers is now in TR project.
(b) More volunteers are now in FR project as compared to ER project.
(c) More volunteers are now in TR project as compared to ER project.
(d) None of the above.

After the withdrawal of volunteers, as indicated in the above question, some new volunteers joined the NGO. Each one of them was allotted only one project in a manner such that, the number of volunteers working in one project alone for each of the three projects became identical. At that point, it was also found that the number of volunteers involved in FR and ER projects was the same as the number of volunteers involved in TR and ER projects. Which of the projects now has the highest number of volunteers?
(a) ER
(b) FR
(c) TR
(d) Cannot be determined

Five boys — A, B, C, D and E — went on a shopping trip. Before shopping, one boy had Rs. 400, one had Rs. 300, two boys had Rs. 200 each and one had Rs. 100. While shopping they did not lend or borrow from each other. After the shopping was over, it was observed that they were left with Rs. 165, Rs. 95, Rs. 70, Rs. 40 and Rs. 10, not necessarily in this order. Further, the following is known about the money they started with, they spent, or they were left with.
I. A started with more money than D.
II. B spent Rs. 15 more than C.
III. E started with more money than just one another person of the group.
IV. A spent the most but did not end with the least.
V. C started with 66.66% of the money that B started with.
VI. D spent the least and ended with more than A and C.
VII. E spent Rs. 35.

Who ended with the maximum amount of money?
a. A
b. B
c. C
d. E

How much money did A spend?
a. Rs. 205
b. Rs. 190
c. Rs. 35
d. Rs. 360

In ascending order of spending, E would rank at which position?
a. 1
b. 2
c. 4
d. 5

Who ended with Rs. 40?
a. A
b. B
c. C
d. D

In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is
(a) 200

(b) 216

(c) 235

(d) 256

There are three cities: A, B and C. Each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly, there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?
a. 6
b. 3
c. 5
d. 10

A shipping clerk has five boxes of different but unknown weights each weighing less than 100 kg. The clerk weights the boxes in pairs. The weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121 kg. What is the weight of the heaviest box?
a. 60 kg
b. 62 kg
c. 64 kg
d. Cannot be determined
e. None of these

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first
stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.
The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage, teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

What is the total number of matches played in the tournament?
a. 28
b. 55
c. 63
d. 35

The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is
a. 5
b. 6
c. 7
d. 4

What is the highest number of wins for a team in the first stage in spite of which it would be eliminated at the end of first stage?
a. 1
b. 2
c. 3
d. 4

What is the number of rounds in the second stage of the tournament?
a. 1
b. 2
c. 3
d. 4

Which of the following statements is true?
a. The winner will have more wins than any other team in the tournament.
b. At the end of the first stage, no team eliminated from the tournament will have more wins than any
of the teams qualifying for the second stage.
c. It is possible that the winner will have the same number of wins in the entire tournament as a
team eliminated at the end of the first stage.
d. The number of teams with exactly one win in the second stage of the tournament is 4.

In my coaching institute there are a total 170 students and they use different vehicles for transportation viz. Bike, Car and Taxi.
The ratio of students using all 3 vehicles to students using at least 2 vehicles is 2 : 9. The ratio of students using only one vehicle to students using at least 2 vehicles is 8 : 9.
Number of students using car only exceeds number of students using bike only by 14.
Number of students using taxi only exceeds number of students using bike only by 12.
Number of students using taxi, bike, car is 90, 93, 97 respectively.

Find: The number of students using all three vehicles; the number of students using no more than one vehicle; the number of students using exactly two vehicles and the number of students who are using both bike and car but not taxi.

A girl leaves home with x flowers, goes to the bank of a nearby river. On the bank of the river, there are four places of worship, standing in a row. She dips all the x flowers into the river. The number of flowers doubles. Then she enters the first place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the second place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the third place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the fourth place of worship, offers y flowers to the deity. Now she is left with no flowers in hand.

If the girl leaves home with 30 flowers, the number of flowers she offers to each deity is
a. 30
b. 31
c. 32
d. 33

The minimum number of flowers that could be offered to each deity is
a. 0
b. 15
c. 16
d. Cannot be determined

The minimum number of flowers with which the girl leaves home is
a. 16
b. 15
c. 0
d. Cannot be determined

Seven university cricket players are to be honoured at a special luncheon. The players will be seated on the dais along one side of a single rectangular table.
I. A and G have to leave the luncheon early and must be seated at the extreme right end of the table, which is closest to the exit.
II. B will receive the Man of the Match Award and must be in the centre chair.
III. C and D who are bitter rivals for the position of wicketkeeper, dislike one another and should be seated as far apart as possible.
IV. E and F are best friends and want to sit together.

Which of the following may not be seated at either end of the table?
a. C
b. D
c. G
d. F

Which of the following pairs may not be seated together?
a. E and A
b. B and D
c. C and F
d. G and D

A survey is conducted in a school among 150 students and it was found that 100 students play cricket, 90 students play football and 80 students play hockey. As physical education was given emphasis in the school, every student had to play at least one sport.
What can be the maximum number of students who play exactly two sports?
What can be the maximum number of students who play exactly one sport?

Substitute different digits (0, 1, 2, …, 9) for different letters in the problem below, so that the corresponding addition is correct and it results in the maximum possible value of MONEY.

P A Y + M E + R E A L = M O N E Y

A very long hallway has 1000 doors numbered 1 to 1000; all doors are initially closed. One by one, 1000 people get into the hall; the first person opens each door, the second closes all door with even numbers, the third person closes door 3, opens door 6, closes door 9, opens door 12 etc. That is, the nth person changes all doors whose numbers are divisible by n. After all 1000 people have got into the hall, find out how many doors will remain open.

A team is to be selected from five men (A, B, C, D and E) and six women (L, M, N, O, P and Q), where A, B and N are lecturers; C, D, L, M and O are engineers, and rest are doctors. The team should be selected subject to the following conditions.
I. If N or D is selected, B should not be selected.
II. When either L or P is selected, the other has to be selected.
III. If any of A, L, or Q is selected, all have to be selected.
IV. D and L cannot be together in a team.
V. If E is selected, M has to be selected and vice versa.
VI. L cannot be with O.

If the team consists of one lecturer, two engineers and three doctors, the members of the team are
a. BELMPA
b. ALEDPQ
c. AELMPQ
d. ADEMPQ

If the team consists of two male lecturers, two lady doctors and one engineer, the members of the team are
a. ABLPQ
b. ABLEQ
c. AQBLO
d. ABLOP

If the team consists of two lecturers, two engineers, two doctors and not more than three women, the members of the team will be one of the following choices :
a. ABELPQ
b. ABCLPQ
c. ABCLMQ
d. ABELNQ

Seven faculty members at a management institute frequent a lounge for strong coffee and stimulating conservation. On being asked about their visit to the lounge last Friday we got the following response:

JC:
I came in first, and the next two persons to enter were SS and SM. When I left the lounge, JP and VR were present in the lounge. DG left with me.

JP:
When I entered the lounge with VR, JC was sitting there. There was someone else, but I cannot remember who it was.

SM:
I went to the lounge for a short while, and met JC, SS, and Dg in the lounge that day.

SS:
I left immediately after SM left.

DG:
I met JC, SS, SM, JP, and VR during my first visit to the lounge. I went back to my office with JC. When I went to the lounge for the second time, JP and VR were there.

PK:
I has some urgent work, so I did not sit in the lounge that day, but just collected my coffee and left. JP and DG were the only people in the lounge while I was there.

VR:
No comments.

Based on the responses, which of the two, JP or DG, entered the lounge first?
(a) JP
(b) DG
(c) Both entered together
(d) Cannot be deduced

Who was sitting with JC when JP entered the lounge?
(a) SS
(b) SM
(c) DG
(d) PK

How many of the seven members did VR meet on Friday in the lounge?
(a) 2
(b) 3
(c) 4
(d) 5

Who were the last two faculty members to leave the lounge?
(a) JC and DG
(b) PK and DG
(c) JP and PK
(d) JP and DG

There are seven coconut trees in Baghban’s garden. Baghban has named his trees as Amitabh, Hrithik, Chunky, Dilip, Fardeen, Feroz and Govinda. The trees are standing in increasing order of their heights, which is not the same as the above order of names. It is known that their heights (in feet) are seven consecutive integral values between 1 and 10 (both inclusive). Further, following clues are given about their positions.

I. Amitabh is 3 ft taller than Dilip.

II. Hrithik stands in the middle of the row of seven trees.

III. The difference in the heights of Feroz and Hrithik, Feroz being shorter, is same as the difference

in the heights of Chunky and Dilip, Chunky being taller.

IV. Feroz is shorter than Govinda.

Difference in heights of Fardeen and Hrithik is same as the difference between the heights of Dilip and which tree?

a. Amitabh

b. Hrithik

c. Chunky

d. Fardeen

Difference in heights of Govinda and Dilip (in inches) is

a. 12

b. 24

c. 48

d. Cannot be determined

The greatest possible height of Amitabh is greater than the least possible height of Feroz by

a. 5 ft

b. 6 ft

c. 7 ft

d. 8 ft

What is the greatest possible height of Amitabh?

a. 7 ft

b. 9 ft

c. 8 ft

d. Cannot be determined

Ten coins are distributed among four people P, Q, R and S such that one of them gets one coin, another gets two coins, the third gets three coins and the fourth gets four coins. It is known that Q gets more coins than P, and S gets fewer coins than R.

If the number of coins distributed to Q is twice the number distributed to P, then which one of the
following is necessarily true?
a. R gets an even number of coins.
b. R gets an odd number of coins.
c. S gets an even number of coins.
d. S gets an odd number of coins.

If R gets at least two more coins than S, then which one of the following is necessarily true?
a. Q gets at least two more coins than S.
b. Q gets more coins than S.
c. P gets more coins than S.
d. P and Q together get at least five coins.

If Q gets fewer coins than R, then which one of the following is not necessarily true?
a. P and Q together get at least four coins.
b. Q and S together get at least four coins.
c. R and S together get at least five coins.
d. P and R together get at least five coins.

A, B, C, D, E and F are a group of friends from a club. There are two housewives, one lecturer, one architect, one accountant and one lawyer in the group. There are two married couples in the group. The lawyer is married to D who is a housewife. No lady in the group is either an architect or an accountant. C, the accountant, is married to F who is a lecturer. A is married to D and E is not a housewife.

What is the profession of E?
a. Lawyer
b. Architect
c. Lecturer
d. Accountant

How many members of the group are male?
a. 2
b. 3
c. 4
d. None of these

Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over person-to-person phone calls so that eventually each person knows all six secrets. None of the Frenchmen knows English, and only one Englishman knows French. What is the minimum number of phone calls needed for the above purpose?
(a) 5
(b) 10
(c) 9
(d) 15

Three labelled boxes containing red and white cricket balls are all mislabelled. It is known that one of the boxes contains only white balls and another one contains only red balls. The third contains a mixture of red and white balls. You are required to correctly label the boxes with the labels red, white and red and white by picking a sample of one ball from only one box. What is the label on the box you should sample?
a. white
b. red
c. red and white
d. Not possible to determine from a sample of one ball

Abraham, Border, Charlie, Dennis and Elmer, and their respective wives recently dined together and were seated at a circular table. The seats were so arranged that men and women alternated and each woman was three places away from her husband. Mrs Charlie sat to the left of Mr Abraham. Mrs Elmer sat two places to the right of Mrs Border. Who sat to the right of Mr Abraham?
a. Mrs Dennis
b. Mrs Elmer
c. Mrs Border
d. Mrs Border or Mrs Dennis

Each digit, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter A, B, C, D, E, F, G, H and I but not necessarily in this order. Further, each of A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
Determine the value of each of the alphabets such that the above conditions are satisfied.

Three people A, B, C are standing in a line in such a way that C can see both A and B. B can see only A, and A can see none of them. They are shown 5 hats, 2 of which are black and 3 of which are white. After this, they are blindfolded and a hat is placed on each of their heads and the blindfolds are removed. None of them can see his own hat. When C is asked regarding the colour of the hat he is wearing, he is unable to answer. Then B is asked and he too is unable to answer but finally A is asked the same question and he gives the colour of his hat. What is the colour of the hat on A’s head?

In a street there are 5 houses painted 5 different colours. In each house lives a person of a different nationality. The 5 homeowners each drink a different beverage, smoke a different brand of cigar, and keep a different pet.
1. The Brit lives in the Red house.
2. The Swede has a Dog.
3. The Dane drinks Tea.
4. The Green house is on the left of the White house.
5. The owner of the Green house drinks Coffee.
6. The person who smokes Pall Mall has Birds.
7. The owner of the Yellow house smokes Dunhills.
8. The man living in the center house drinks Milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blends lives next to the one who has Cats.
11. The man who has Horses lives next to the man who smokes Dunhills.
12. The man who smokes Blue Master drinks Beer.
13. The German smokes Prince.
14. The Norwegian lives next to the Blue house.
15. The man who smokes Blends has a neighbour who drinks water.

Find out the correct combination of drinks, pets, smokes and colour of the house for each of the five persons.