Puzzle Set II

Four couples are going to the movie. Each row holds eight seats. Betty and Jim don't want to sit next to Alice and Tom. Alice and Tom don't want to sit next to Gertrude and Bill. On the otherhand, Sally and Bob don't want to sit next to Betty and Jim.

How can the couples arrange themselves in a row so that they all sit where they would like?
Submitted by : Tara Smith

Answer

From the given data, it can be inferred that:
(Sally & Bob) NOT (Betty & Jim) NOT (Alice & Tom) NOT (Gertrude & Bill)

(A) NOT (B) means A and B can not seat next to each other.

Now, it is obvious that (Betty & Jim) and (Alice & Tom) will occupy the corner seats as both of them can have only one neighbour. Therefore,
(Gertrude & Bill) will seat next to (Betty & Jim)
(Sally & Bob) will seat next to (Gertrude & Bill)
(Alice & Tom) will seat next to (Sally & Bob)

Thus, there are two possible arrangements - a mirror images of each other.

1. (Betty & Jim) - (Gertrude & Bill) - (Sally & Bob) - (Alice & Tom)
2. (Alice & Tom) - (Sally & Bob) - (Gertrude & Bill) - (Betty & Jim)
Substitute digits for the letters to make the following addition problem true.
W H O S E

T E E T H

A R E

+ A S

-------------------

S W O R D S
Note that the leftmost letter can't be zero in any word. Also, there must be a one-to-one mapping between digits and letters. e.g. if you substitute 3 for the letter H, no other letter can be 3 and all other H in the puzzle must be 3.
Answer

It is obvious that S=1 and T=9.

Also, (H + E) should be greater than 10 and hence, (E + H + E) must 20. Thus, there are 3 possible values for (E, H) pair: (6, 8) or (7, 6) or (8, 4). Use trial-n-error and everything will fit-in.

W H O S E 2 8 5 1 6

T E E T H 9 6 6 9 8

A R E 4 7 6

+ A S + 4 1

------------------- -------------------

S W O R D S 1 2 5 7 3 1

When Socrates was imprisoned for being a disturbing influence, he was held in high esteem by his guards. All four of them hoped that something would occur that would facilitate his escape. One evening, the guard who was on duty intentionally left the cell door open so that Socrates could leave for distant parts.

Socrates did not attempt to escape, as it was his philosophy that if you accept society's rules, you must also accept it's punishments. However, the open door was considered by the authorities to be a serious matter. It was not clear which guard was on that evening. The four guards make the following statements in their defense:

Aaron:
A) I did not leave the door open.
B) Clement was the one who did it.

Bob:
A) I was not the one who was on duty that evening.
B) Aaron was on duty.

Clement:
A) Bob was the one on duty that evening.
B) I hoped Socrates would escape.

David:
A) I did not leave the door open.
B) I was not surprised that Socrates did not attempt to escape.

Considering that, in total, three statements are true, and five statements are false, which guard is guiltyAnswer

David is the guilty.

Note that "All four of them hoped that something would occur that would facilitate his escape". It makes Clement's statement B True and David's statement B False.

Now consider each of them as a guilty, one at a time.
Aaron Bob Clement David True
Stmts
A B A B A B A B
If Aaron is guilty False False True True False True True False 4
If Bob is guilty True False False False True True True False 4
If Clement is guilty True True True False False True True False 5
If David is guilty True False True False False True False False 3

Since in total, three statements are true and five statements are false. It is clear from the above table that David is?
Brain Teaser No : 00474

Given any whole number take the sum of the digits, and the product of the digits, and multiply these together to get a new whole number.

For example, starting with 6712, the sum of the digits is (6+7+1+2) = 16, and the product of the digits is (6*7*1*2) = 84. The answer in this case is then 84 x 16 = 1344.

If we do this again starting from 1344, we get (1+3+4+4) * (1*3*4*4) = 576

And yet again (5+7+6) * (5*7*6) = 3780

At this stage we know what the next answer will be (without working it out) because, as one digit is 0, the product of the digits will be 0, and hence the answer will also be 0.

Can you find any numbers to which when we apply the above mentioned rule repeatedly, we never end up at 0?
Brain Teaser No : 00474

Given any whole number take the sum of the digits, and the product of the digits, and multiply these together to get a new whole number.

For example, starting with 6712, the sum of the digits is (6+7+1+2) = 16, and the product of the digits is (6*7*1*2) = 84. The answer in this case is then 84 x 16 = 1344.

If we do this again starting from 1344, we get (1+3+4+4) * (1*3*4*4) = 576

And yet again (5+7+6) * (5*7*6) = 3780

At this stage we know what the next answer will be (without working it out) because, as one digit is 0, the product of the digits will be 0, and hence the answer will also be 0.

Can you find any numbers to which when we apply the above mentioned rule repeatedly, we never end up at 0?

There were N stations on a railroad. After adding X stations 46 additional tickets have to be printed.

Find N and X.
Answer

Let before adding X stations, total number of tickets
t = N(N-1)

After adding X stations total number of tickets are
t + 46 = (N+X)(N+X-1)

Subtracting 1st from 2nd
46 = (N+X)(N+X-1) - N(N-1)
46 = N2 + NX - N + NX + X2 - X - N2 + N
46 = 2NX + X2 - X
46 = (2N - 1)X + X2
X2 + (2N - 1)X - 46 = 0

Now there are only two possible factors of 46. They are (46,1) and (23,2)

Case I: (46,1)
2N - 1 = 45
2N = 46
N = 23
And X = 1

Case II: (23,2)
2N - 1 = 21
2N = 22
N = 11
And X = 2

Hence, there are 2 possible answers.

An emergency vehicle travels 10 miles at a speed of 50 miles per hour.

How fast must the vehicle travel on the return trip if the round-trip travel time is to be 20 minutes?
Answer

75 miles per hour

While going to the destination, the vehicle travels 10 mils at the speed of 50 miles per hour. So the time taken to travel 10 miles is
= (60 * 10) / 50
= 12 minutes

Now it's given that round-trip travel time is 20 minutes. So the vehicle should complete its return trip of 10 miles in 8 minutes. So the speed of the vehicle must
= (60 * 10) / 8
= 75 miles per hour
All of the students at a college are majoring in psychology, business, or both. 73% of the students are psychology majors, & 62% are business majors.

If there are 200 students, how many of them are majoring in both psychology & business?
Answer

70 students are majoring in both, psychology & business

If 73% of the students are psychology majors, we know that 27% are not psychology majors. By the same reasoning, 38% are not business majors, because 62% of the students do major in business. So: 27 + 38 = 65

65% of the students are not majoring in both psychology & business, so 35% are double majors, a total of 70 students.
Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination 1hr and 4hrs respectively after passing each other.

Answer


The speed of Bangalore-Mysore train is TWICE the speed of Mysore-Bangalore train.

Let the distance between Bangalore and Mysore is D kms.
Also, let speed of the train from Bangalore to Mysore is X km/hr and speed of the tain from Mysore to Bangalore is Y km/hr.

Now, assume that both the trains met each other at T kms from the Bangalore (point P in figure)
Time taken by Bangalore-Mysore train to reach P = Time taken by Mysore-Bangalore train to reach P
( T / X ) = ( D - T ) / Y -----equ(I)

Also, Bangalore-Mysore train and Mysore-Bangalore train arrive destination 1 hr and 4 hrs respectively after passing each other. It means that Bangalore-Mysore train travels (D - T) kms in 1 hr at X km/hr and Mysore-Bangalore train travels T kms in 4 hrs at Y km/hr. Hence,
( D - T ) = X and
T = 4 * Y

Substituting these values in equation I, we get
( 4 * Y ) / X = X / Y
4 * Y * Y = X * X
2 * Y = X

Hence, the speed of Bangalore-Mysore train is TWICE the speed of Mysore-Bangalore train.How much faster is one train from other?

Answer

49 times

Let's assume that everyone clinked their mug with friend to his left only. It means that there are total 49 clinks. Now the right clink of each person is left clink of the person on right which is already happened. Hence, there are only 49 clinks.

Mrs. Watsherface had a garage sale. A custmer named Gina bought an old lamp and a rug. She paid a total of $5.25 for everything. The rug cost 25 cents more than the lamp.

How much did each cost?
Submitted by : Kimi
Answer

The lamp cost $ 2.50 and the rug cost $ 2.75

A simple one.

Assume that the lamp cost $ L.
Hence the rug must have cost $ (L + 0.25)
Also, total cost is $ 5.25, Hence the equation :
L + L + 0.25 = 5.25
2 * L = 5
L = 2.50

Hence, the lamp cost $ 2.50 and the rug cost $ 2.75

Brain Teaser No : 00518

Write 1111......(243 times) i.e. a 243 digit number with all 1s.

Prove that it is divisible by 243.
SubmittAnswer

Prove it using the mathematical induction.

First here are a couple of things to note:

[1] A number whose digits add up to a multiple of three is divisable by 3.
e.g. 369: 3+6+9=18: 1+8=9 which is a multiple of 3 hence 369 is divisable by 3.

[2] Whenever a number (X) is multiplied with another number (Y) then the product (X*Y) will have all the factors of X as well as all the factors of Y in its set of factors.
e.g. if X has factors of (1,P,Q,X) and Y has factors of (1,Q,R,Y) then X*Y has factors of (1,P,Q,Q,R,X,Y).


Let
N = any series of digits (e.g. N=369)
D = the number of digits in N (e.g. if N=369 then D=3)
P = is a number constructed in the following way : a 1, followed by (D-1) 0s, followed by another 1, followed by (D-1) 0s, followed by another 1. (e.g. if N=369 then D=3 and P would be 1001001) Note that P will always be divisible by 3.

Also, if we multiply N with P we are essentially repeating N for (D-1) times.
e.g. if N=369 then D=3, P=1001001 and N*P=369369369

Let's start with N=111. It is clear that N is divisible by 3. (From [1])
Also, D=3 and P=1001001
N*P=111111111 (9 times)
The resulting number 111111111 must be divisible by 9 as N and P both are divisible by 3.


Now, let's start with N=111111111. It is clear that N is divisible by 9.
Also, D=9 and P=1000000001000000001
N*P=111111111... (27 times)
The resulting number 1111111... (27 times) must be divisible by 27 as N is divisible by 9 and P is divisible by 3.

Repeat the same procedure for N=1111111... (27 times) The resulting number 1111111... (81 times) must be divisible by 81 as N is divisible by 27 and P is divisible by 3.

Similarly, for N=1111111... (81 times) The resulting number 1111111... (243 times) must be divisible by 243 as N is divisible by 81 and P is divisible by 3.

Thus, 1111111... (243 times) is divisible by 243.

Thanks to Ryan Hutcherson for solution !!!
edKaran bought a little box of midget matches, each one inch in length. He found that he could arrange them all in the form of a triangle whose area was just as many square inches as there were matches.

He then used up six of the matches, and found that with the remainder he could again construct another triangle whose area was just as many square inches as there were matches.

And using another six matches he could again do precisely the same.

How many matches were there in the box originally?

Note that the match-box can hold maximum of 50 matches.
Answer

Initially, there were 42 or 36 matches in the match-box.

There are 42 matches in the box with which he could form a triangle 20, 15, 7, with an area of 42 square inches. After 6 matches had been used, the remaining 36 matches would form a triangle 17, 10, 9, with an area of 36 square inches. After using another 6 matches, the remaining 30 matches would form a triangle 13, 12, 5, with an area of 30 square inches. After using another 6, the 24 remaining would form a triangle 10, 8, 6, with an area of 24 square inches.

Thus, there are two possible answers. There were either 42 or 36 matches in the match-box.


Also it is interesting to know that there are just 5 such triangles for which the perimeter and the area is the same (assuming all sides are integers) and they are :
1. 24 (10, 8, 6)
2. 30 (13, 12, 5)
3. 36 (17, 10, 9)
4. 42 (20, 15, 7)
5. 60 (29, 25, 6)

Find the values of each of the alphabets.

N O O N

S O O N

+ M O O N

----------

J U N E
Answer

Using trial and error. There are 2 solutions to it and may be more.

2 4 4 2

1 4 4 2

+ 5 4 4 2

----------

9 3 2 6


4 1 1 4

5 1 1 4

+ 0 1 1 4

----------

9 3 4 2

We have to fill number from 1 to 12 at the intersection point of two or more lines. We have to construct a star using two triangle. The sum of all number lying in straight lines should be same. This can be easilty understood by the fig. and hence solved.
Submitted by : Vaibhav Gupta
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We have one answer where sum of all the numbers lying in straight lines is 26.

If you have others, do submit them.
Brain Teaser No : 00355

Montu, Bantu, Chantu and Pintu have pets.

Montu says, "If Pintu and I each have a dog, then exactly one of Bantu and Chantu has a dog."

Bantu says, "If Chantu and I each have a cat, then exactly one of Montu and Pintu has a dog."

Chantu says, "If Montu and I each have a dog, then exactly one of Bantu and Pintu has a cat."

Pintu says, "If Bantu and I each have a cat, then exactly one of Bantu and I has a dog."

Only one of the four is telling the truth. Who is telling the truth?

Answer

Bantu is telling the truth.

For a IF-THEN statement to be false, IF part has to be true and THEN part has to be false.

Since only one statement is true and remaining three are false, IF part of three statements are true & THEN part of one statement is true. Let's put the given information in table. The pet-name in the normal text represents the IF part and the pet-name in round brackets represents the THEN part.
Montu Bantu Chantu Pintu
Montu says Dog (Dog) (Dog) Dog
Bantu says (Dog) Cat Cat (Dog)
Chantu says Dog (Cat) Dog (Cat)
Pintu says Cat
(Dog) Cat
(Dog)

It is clear that the IF part of the statements made by Montu, Chantu and Pintu are true as they do not contradict each other. And the IF part of the statement made by Bantu is false.

Thus, Bantu is telling the truth.

Montu have a Dog and may or may not have a Cat.
Bantu have a Cat.
Chantu have a Dog.
Pintu have a Dog and a Cat.