The Domino on The Chessboard Challenge

There is an 8 by 8 chessboard in which two diagonally opposite corners have been cut off.You are given 31 dominos, and a single domino can cover exactly two squares. Can you use the 31 dominos to cover the entire board?

Simple Arrangement? Check out it's possibility!

Impossible Dominos' Arrangement on Chessboard

What was the challenge given?

Initial mathematical calculations might suggest that the task is pretty simple. If 2 square are cut off from 64 squares then 62 squares will be left which are enough for 31 dominos (each covering 2 squares).

But, that is not the case. Since, 2 diagonally opposite squares are removed, they has to be either black or white like shown below with shaded regions.

We need 1 black and 1 White square for placement of 1 domino on the chessboard.That is 31 Black and 31 White squares are needed to give cover for 31 dominos.

 In the above 2 cases, there are either 32 Black and 30 White or 30 Black and 32 White squares are available.

This makes the task of placing 31 dominos on the chessboard (whose 2 diagonally opposite squares are removed) impossible! 

The Ping Pong Puzzle

Three friends (A, B and C) are playing ping pong. They play the usual way: the winner stays on, and the loser waits his/her turn again. At the end of the day, they summarize the number of games that each of them played:

A played 10
B played 15
C played 17.

Who lost the second game? 

This person played & lost the second game! 

Participant of the Second Game!

How games were played?

A played 10, B played 15 and C played 17 games. So total number of presences are 10 + 15 + 17 = 42. Every 2 presences form a game. Hence, the number of games played are 42/2 = 21.

Let's take into consideration the minimum number of games that a player can play. For that, he need to loose every game that he has played. That is, if he has played first game then he must have out in second but replaced looser of second in third game. In short, he must have played odd numbered of games like 1,3,5,7,9,11,13,15,17,19,21.That's 11 games in total.

And if he had made debut in second game then he must had played even numbered games like 2,4,6,8,10,12,14,16,18,20. That's 10 games in total.

Since, in the case only A has played 10 games, he must have made debut in second game where he lost that game to make comeback in 4th game thereby replacing looser of third game.

Maximum Runs That Batsman Can Score?

In a one day international cricket match, considering no extras(no wides, no ‘no’ balls, etc.) and no overthrows.

What is the maximum number of runs that a batsman can score in an ideal case ?

Note: Here we assume ideal and little practical scenario. We assume that batsman can not run for more than 3 runs in a ball, as otherwise there is no limit, he can run infinite runs(theoretically) in a ball, as far as opposite team does not catch the ball.”

Could be tricky! Here is correct number! 

Calculation of Maximum Runs by Batsman

What was the question?

It's not as straight forward as it seems at first glance. That is one might think that the maximum score that one can score by hitting 6 on every ball of 50 overs is 50 x 6 x 6 = 1800. 

No doubt, 1800 can be the maximum team score but not the individual score.Since batsman rotates strike every over, here both batsmen share these 1800 runs as 900 to each.

However, if the batsman on strike runs 3 runs on the last ball of the over then he can hit 5 more sixes in next over as strike is rotated back to him in next over. He can continue in this way till 49th over. And in 50th over he can hit 6 sixes on 6 balls.

Maximum Individual Score = 49 x [(5x6)+3]  + 36 = 1617 + 36 = 1653.

In this case, the batman at the non-striker end scores 0 runs as he doesn't get strike on a single ball.

"Who is telling the truth?"

King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always say the truth.

One day, 4 servants met :

The blue one says: “Altogether we have 28 legs”;

the green one says: “Altogether we have 27 legs”;

the yellow one says: “Altogether we have 26 legs”;

the red one says: “Altogether we have 25 legs”.

 
What is the color of the servant that says the truth?

"I'm telling the truth and my color is..." 

"Listen to me; I'm telling the truth!"

What others making statements?

Since, each of 4 servants telling different numbers then only 1 of them must be telling the truth & other must be lying.

In that case, all 3 must be having 7 legs i.e. 21 legs while the one who is making true statement must have either 6 or 8 legs. Therefore, there must be 21 + 6 = 27 or 21 + 8 = 29 legs altogether.

Nobody is saying that they altogether have 29 legs but the green octopus is saying that they altogether have 27 legs. Hence, the green octopus must be telling the truth. 

An Expensive Pearl Necklace

A pearl necklace has 33 pearls with the largest and most valuable in the middle.
Starting from one end, each successive pearl is worth $100 more than the one before (up to the middle one), but starting from the other end each pearl is worth $150 more than the one before, up to the big pearl. The whole necklace is worth $65 000.

What is the value of the middle pearl?

Here is the calculation of it's cost!

The Cost of Middle Pearl of the Necklace

What is the question?

Let X be the cost of the middle pearl of the necklace.

There are 33 total pearls in the necklace meaning that 16 pearls are on the left and 16 are on the right.

The values of the pearls at the left must be ( X - 100 ), ( X - 200 ),..........( X - 1600 ) and those which are right must have value ( X - 150 ), ( X - 300 ),................( X - 2400 ).

The total value of the pearls at left = ( X - 100 ) + ( X - 200 ) + .......... + ( X - 1600 ) 

                                                    = 16X - 100 ( 1 + 2 + ...... + 16)
 
The total value of the pearls at left = 16X - 100 (136)

The total value of the pearls at right = ( X - 150 ) + ( X - 300 ) +............... + ( X - 2400 )                                                     
                                                      = 16X - 150 ( 1 + 2 + ...... + 16)

The total value of the pearls at right =  16X - 150 (136)  

Therefore,

The total cost of necklace = The total value of the pearls at left + The cost of middle pearl + The total value of the pearls at right 

 The total cost of necklace =  16X - 100 (136) + X + 16X - 150 (136) 
               
                                       = 33X - 250 (136)

The total cost of necklace  = 33X - 34000

But, the total cost of the necklace given is $65000,

The total cost of necklace  = 33X - 34000 = 65000

33X = 99000

X = 3000.

Hence, the cost of the middle pearl is $3000. So the cost of the leftmost pearl is 3000 - 1600 = 1400 and the rightmost pearl is 3000 - 2400 = 600.

Effect of Average Speed on Time

If a car had increased its average speed for a 210 mile journey by 5 mph, the journey would have been completed in one hour less. What was the original speed of the car for the journey?

Here is the calculation of averages speed!

Calculation of the Original Speed!

What was the question?

Let S1 be the original speed and S2 be the modified speed and T1 be the time taken with speed S1 and T2 be the time taken with speed S2.

As per given data,

T1 - T2 = 1 hr.

D/S1 - D/S2 = 1 hr.

Here, D = 210 miles, S2 = S1 + 5,

210/S1 - 210/(S1+5) = 1

210(S1+5) - 210s = 1S1(S1+5)

S1^2 + 5S1 - 1050 = 0

(S1-30)(S1+35) = 0 

S1 = 30 or S1 = -35.

Since speed can't be negative, S1 = 30 mph.

Hence, the original speed is 30 mph and average speed is 30 + 5 = 35 mph.

 
With the original speed it would have taken 210/30 = 7 hours but with average speed it took only 210/35 = 6 hours saving 1 hour of time. 

The Missing Number?

99 unique numbers between 1 and 100 are listed one by one, with 5 seconds pause between every two consecutive numbers. If you are not allowed to take any notes, what is the best way to figure out which is the missing number? 

This is how to correctly guess the missing number!

Guessing The Missing Number!

What was the challenge given?

Just keep up adding the given numbers and remember only the last two digits of the sum.

The sum of all numbers from 1 to 100 is 5050, so if you know the sum of all the listed numbers, you will know the missing number as well. 

At the end, if the result is less than 50, subtract it from 50. If the result is larger than 50, subtract it from 150.