The Lightning Fast Addition!

A  story tells that, as a 10-year-old schoolboy, Carl Friedrich Gauss was asked to find the sum of the first 100 integers. The tyrannical schoolmaster, who had intended this task to occupy the boy for some time, was astonished when Gauss presented the correct answer, 5050, almost immediately.

How did Gauss find it?

Actually, he used this trick! 

 

Trick for The Lightneing Fast Addition!

Why lightning fast speed needed?

Gauss attached 0 to the series and made pairs of numbers having addition 100.

100 + 0 = 100

99 + 1 = 100

98 + 2 = 100

97 + 3 = 100

96 + 4 = 100

95 + 5 = 100
..
..
..
..
..
..
51 + 49 = 100

This way he got 50 pair of integers (ranging in between 1-100) having sum equal to 100.

So sum of these 50 pairs = 100x50 = 5000.

 
And the number 50 left added to above total to get sum of integers 1 - 100 as 5000 + 50 = 5050

Probability of The Correct Answer?

This is a popular probability puzzle in which you have to select the correct answer at random from the four options below.

Can you tell, whats the probability of choosing correct answer in this random manner.

1) 1/4
2) 1/2
3) 1
4) 1/4

And the correct answer is......

Finding The Probability of Correct Answer

What was the question?

It can't be 1/4 as 1/4 appears 2 times in given 4 options as probability of correct answer when random selection of option in that case would be 2/4 = 1/2. This will be contradiction.

It can't be 1/2 either since the probability of the 1 correct answer out of 4 available options on random selection would be 1/4. That will be contradiction again.

It can't be 1 too as again probability of the 1 correct answer out of 4 available options on random selection would be 1/4. Once again this is contradiction.

Hence, the probability of the 1 correct answer out of 4 available options on random selection would be 0.

The Mistimed Clock!

Andrea’s only timepiece is a clock that’s fixed to the wall. One day she forgets to wind it and it stops.

She travels across town to have dinner with a friend whose own clock is always correct. When she returns home, she makes a simple calculation and sets her own clock accurately.

 
How does she manage this without knowing the travel time between her house and her friend’s?


That's how she manages to set it accurately!
 

Correcting The Mistimed Clock!

 How it was mistimed?

Andrea winds her clock & sets it to the arbitrary time. Then, she leaves her house and when she reaches her friend's house, she note down the correct time accurately. Now, after having dinner, she notes down the correct time once again before leaving her friend's house.

After returning to home, she finds her own clock acted as 'timer' for her entire trip. It has counted time that she needed to reach her friend's house + time that she spent at her friend's house + time she needed to return back to home.

Since, Andrea had noted timings at which she reached & left her friend's house, she can calculate the time she spent at her friend's house. After subtracting this time duration from her unique timer count she gets the time she needed to reach to & return from her friend's house.

She must have taken the same time to travel from her home to her friend's home and her friend's home to her home. So dividing the count after subtracting 'stay time' she can get how much time she needed to return back to home.

Since, she had noted correct time when she left her friend's home, now by adding time that she needed to return back to home to that, she sets her own clock accurately with correct time.

Let's try to understand it with example.

Suppose she sets her own clock at 12:00 o' clock and leave her house. Suppose she reaches her friend's home and note down the correct time as 3:00 PM. After having dinner she leaves friend's home at 4:00 PM.

After returning back to home she finds her own clock showing say 2:00 PM. That means, she spent 120 minutes outside her home with includes time of travel to and from friends home along with time for which she spent with her friend. If time of stay at her friend is subtracted from above count, then it's clear that she needed 60 minutes to travel to & return back from friend's home.

That is, she needed 30 minutes for travel the distance between 2 homes. Since, she had noted correct time as 3:00 PM when she left friend's home, she can set her own clock accurately at 3:30 PM.

The Unfair Arrangement!

Andy and Bill are traveling when they meet Carl. Andy has 5 loaves of bread and Bill has 3; Carl has none and asks to share theirs, promising to pay them 8 gold pieces when they reach the next town.

They agree and divide the bread equally among them. When they reach the next town, Carl offers 5 gold pieces to Andy and 3 to Bill.

“Excuse me,” says Andy. “That’s not equitable.” He proposes another arrangement, which, on consideration, Bill and Carl agree is correct and fair.

How do they divide the 8 gold pieces?

This is fair arrangement of gold distribution! 

Source 

Correcting The Unfair Arrangement!

How unfair the arrangement was?

First we need to know how 8 loaves (5 of Andy & 3 of Bill) are equally distributes among 3.

If each of them is cut into 2 parts then total 16 loaves would be there which can't be divided equally among 3.

Suppose, each of loaves is divided into 3 parts making total 24 loaves available.

Now, Andy makes 15 pieces of his 5 loaves. He eats 8 and gives the remaining 7 to Carl.

Bill makes 9 pieces of his 3 loaves. He eats 8 and gives the remaining 1 to Carl.

This way, Carl too gets 8 pieces and 8 breads are distributed equally among 3.

 
Obviously, Carl should pay 7 gold pieces to Andy for his 7 pieces and 1 gold piece to Bill for the only piece offered by Bill. 
 

Breaking The Safe in 5 Minutes?

Charlie Croker and his team need to break the safe to finish a secret job named "Italian Job" in exactly a five minutes.

They got just one chance and five minutes to finish the job else the local police will be informed.

He got following clues

1st Clue: Exactly one number is perfectly placed.
9 2 5

2nd clue: Everything is incorrect.
9 3 8

3rd clue: Two numbers are part of the code of the safe, but are wrongly placed.
4 9 6

4th clue: One number is part of the code of the safe, but is wrongly placed.
5 8 1

5th clue: One number is part of the code of the safe, but is wrongly placed.
1 2 6 

This should be the process!

To Break The Safe in 5 Minutes...

What was the challenge? 

Re listing all the clues...

1st Clue: Exactly one number is perfectly placed.
9 2 5

2nd clue: Everything is incorrect.
9 3 8

3rd clue: Two numbers are part of the code of the safe, but are wrongly placed.
4 9 6

4th clue: One number is part of the code of the safe, but is wrongly placed.
5 8 1

5th clue: One number is part of the code of the safe, but is wrongly placed.
1 2 6 

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

From 2nd clue, it's clear that 9,3,8 are not part of the code.

Hence correct number suggested by 1st clue must be 2 or 5

Since 9 is not part of the code, the other 2 correct numbers that 3rd clue pointing must be 4 and 6.

If 6 is the part of code, then 1 & 2 are not as 5th clue is suggesting.

And since 2 isn't part of the code then 1st clue must be pointing 5 is correct digit placed in right position.

The 4th clue is also suggesting that 5 is the part of the code but not 1 or 8.

If 5 is correct at it's position as per first clue then 4 and 6 must be occupying other 2 places.

As per 3rd clue position of 4 is wrong, it must be at second place and hence 6 at first place.

Hence the code is 645!

The Game Of Guesses

In a contest, four fruits (an apple, a banana, an orange, and a pear) have been placed in four closed boxes (one fruit per box). People may guess which fruit is in which box. 123 people participate in the contest. When the boxes are opened, it turns out that 43 people have guessed none of the fruits correctly, 39 people have guessed one fruit correctly, and 31 people have guessed two fruits correctly.

How many people have guessed three fruits correctly, and how many people have guessed four fruits correctly?

Escape to answer without getting tricked! 

Correct Guesses From The Game Of Guesses

What was the game?

There is absolutely no way that somebody has guessed 3 correctly since if 3 are correct then 4th has to be correct. Hence, nobody guessed 3 correctly.

So number of people with all 4 guess correct is equal to 123 - 43 - 39 - 31 = 10.

10 people guessed all the 4 fruits correctly.

Test of an Examiner

Five students - Adam, Cabe, Justin, Michael and Vince appeared for a competitive exam. There were total five questions asked from them from which were two multiple choice questions (a, b or c) and three were true/false questions. Their answers are given as follows:

Name I II III IV V

Cabe c b True True False

Adam c c True True True

Justin a c False True True

Michael b a True True False

Vince b c True False True

Also, no two students got the same number of correct answers. Can you tell the correct answer? Also, what are their individual score?

Know correct answers and scores here!

Source 

Responding To Test of an Examiner

What was the test?

There are 2 possibilities of scores & that are either 0,1,2,3,4 or 1,2,3,4,5. First of all, let's arrange students' responses in order like below.

Table 1

What we notice here is that, there are few responses to same question by different student matching.

For the Question III, only Justin given different answers than other.

Case 1 : If we assume Justin's answer is correct then rest of all are wrong in response to Question III. That means either maximum score in test is 4 or Justin himself has scored 1 to 5.

Let's test that apart from Justin who can have score of 4. If any body other scores 4 then he must share at least 3 similar answers with other (excluding Answer III; refer image below). Only Adam has exact 3 matching responses with Justin.

Table 2

If Adam's score is 4 (Answers to I, II, IV, V are correct) then, Justin too would score 4 (Answers to II, III, IV,V are correct) since Adam & Justin have same responses to Questions II, IV,V).
  
If nobody scoring as 4 then Justin can have score of 4 or 5.

Case 1.1 : If his score is 4 then there has to be somebody has to be there scoring 0. Now Vince and Adam has at least 2 responses matching with the Justin. That means they can't score 0 since even 1 answer is wrong as Justin the other must be correct as Justin. Michael or Cabe can have 0 score in the case. If anybody of them has score 0 then answer as a TRUE to the Question IV is incorrect i.e. correct Answer IV is FALSE. So Justin is WRONG in Answer IV only. In short, a, c, FALSE, FALSE, TRUE is correct combination of answers. But thing is here in the case both Michael and Cabe would have score 0! Hence Justin's score can't be 4 too.

Case 1.2 : If Justin's score is 5, then a, c, FALSE, TRUE, TRUE are the right answers. No one would score 4 in that case with 3 as second highest by Adam.

Wrong Looking Correct Mathematical Equation!

The following question it puts forth you:

25 - 55 + (85 + 65) = ?

Then, you are told that even though you might think its wrong, the correct answer is actually 5!

Whats your reaction to it? How can this be true? 

 That's how it's perfectly correct!

That's How Equation is Correct!

Why it was looking wrong? 

If you read the data carefully then you will notice '!' attached to number 5 which is being claimed answer. Actually claimed answer is 5! not 5 Read it again...

"Then, you are told that even though you might think its wrong, the correct answer is actually 5!."

Now use of '!' is not limited to the sentences only. In mathematics it's a 'factorial'.

So 5! = 5 x 4 x 3 x 2 x 1 = 120 and 25 - 55 + (85 + 65) = 120 and hence,

25 - 55 + (85 + 65) = 5! 

Now doesn't it look the correct equation? 

How Accurate You are?

In a competitive exam, each correct answer could win you 10 points and each wrong answer could lose you 5 points. You sat in the exam and answered all the 20 questions, which were given in the exam.

When you checked the result, you had scored 125 marks in the test.
Can you calculate how many answers given by you were correct and wrong ?

These should be those numbers! 

Source 

  

Analysis Of Your Result

What was the test ?

Let C be the number of correct answers and W be the number of wrong answers.

Since there are 20 question in total,

C + W = 20    .....(1)

and the score 125 must be subtraction of marks obtained for correct answer and marks due to wrong answers.

10C - 5W = 125   .....(2)

Multiplying (1) by 5 and then adding it to (2),

5C + 5W + 10C - 5W = 100 + 125

15C = 225

C = 15.

From (1), W = 20 - C = 20 - 5 = 5.

Hence, your 15 answers are correct while 5 answers are wrong.

Just Try To Crack It!

Can you tell the correct key?

Here is the step-by-step process!

Source 

Cracking of The Code in Steps...

What was the challenge?

Let's number the clues as 1, 2, 3.

Now following step by step process here onward. 

1. The numbers 3 & 1 are common in first & third combinations. Now both must not be the part of original number as in that case Clue 1 will be invalid. 

2. If numbers 3 & 1 are not correct in third combination then the correct 2 numbers must be among 5,7,9.

3. But it can't be both 7 and 9 as again that would make clue 1 invalid! Hence, the 5 is part of the original key & in correct position as in third combination. So we have got first digit of key as a 5.

4. The 1 correct number in clue 2 is 5 & that's in wrong position. If other is assumed to be 9 & to be in right position then it contradict the clue 3. So the second digit must be 7.

5.The only correct number in clue 1 is 7 & that's in wrong position. That means numbers 1,3,4,9 must not be the part of the code.

6. Since 3,4,9 eliminated in previous steps, the only number that is correct and in right position must be 6 in suggestion made by clue 2. So far we have got 3 digits of the code as 576XX.

7. Last 2 digits can be any combination from 0,2,5,7,6,8. Now addition of all digits is equal to the number formed by last 2 digits. It's impossible that the addition of all digits exceeds 50. Hence, the second last digit must be 2.

8. Now both 57620 or 57628 are perfectly valid where sum of all digits equals to number formed by last 2 digits.