Freedom From The Hell !

How you are trapped in hell?

Alekhine always plays black and Nimzowitsch plays always white. Obviously, you will be forced to choose white against Alekhine and black against Nimzowitsch. 

Wait for Nimzowitsch's first move and then play the same move on Alekhine’s board. Note how Alekhine responds to move & copy that move on Nimzowitsch's board.

This way, effectively Nimzowitsch is playing against Alekhine and you are just transferring moves between to 2 masters.

Since, they never lost a single chess game, there are high chances that the game between two ends in draw.

Even if Alekhine wins with his black then you are winning against Nimzowitsch as you are copying Alekhine's move against Nimzowitsch's white using your black. Same is case if Nimzowitsch wins.

In short, you will end up with either draw with both or win against at least one (against both is impossible). You will be free in any case.

And in fact, you need not to have knowledge of how to play chess to get freedom from the hell.

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.

Dropping Height For Egg Breakdown!

There is a building of 100 floors.

-If an egg drops from the Nth floor or above it will break.

-If it’s dropped from any floor below, it will not break.

You’re given 2 eggs.

Find N.

How many drops you need to make?

What strategy should you adopt to minimize the number egg drops it takes to find the solution?

'This' should be the strategy! 

Finding The Dropping Height For Egg Breakdown

What was the task?

Let's suppose, the we drop the egg from N the floor & it breaks then we need to do linear search till (N-1) th floor. For example, we dropped egg from 10 the floor & it breaks then we need to test 1-9 floors using 2nd egg to find the exact floor from which the egg breaks on dropping. So the best case would be 10 drops if the first egg breaks when dropped from 10 the floor.But that's only possible if the egg breaks on drop from 10th floor.

The worst case would be when egg doesn't break at 10,20,30,40,50,60,70,80,90 and breaks at 100th floor. Here, once again 91-99 need to be tested using other egg to find exact floor from which egg will break on drop.So in worst case, 19 drops would be needed.

Best way to minimize the number of drops required is to minimize the linear search (91-99 above  in worst case) that we need to do with the second egg after 1st egg breaks on drop from particular floor (100th floor in above worst case).

So after dropping egg from N th floor & if egg doesn't break then instead of going to next Nth floor, better to go N + (N-1) th floor. And now if the egg break here at N + (N-1) th floor then we need to do linear search from (N+1) th floor to N + (N-1) th floor instead of (N+1) th floor to (N + N) th floor. That's 1 less linear search than that needed if we go to the next N th floor if egg doesn't break on drop from Nth floor.

After dropping egg from N + (N-1) th floor, if it doesn't break then we should go the the N + (N-1) + (N-2) th floor.

Adding all instances of drops i.e. drop at Nth, drop at N + (N-1) th , drop at N + (N-1) + (N-2) & so on gives us

N + (N-1) + (N-2) + (N-3).......+1 = N(N+1)/2.

which shouldn't exceed 100 as there are only 100 floors & hence total number of drops must not be greater than 100.

So,

N(N+1)/2 >= 100

N^2 + N - 200 >= 0

This is

This is quadratic equation in form ax^2 + bx + c = 0 where x = [-b +- (b^2 -4ac)^0.5]/2a.

Solving above for N gives,

N = 13.651

Rounding value of N to 14 in the case.

So we should start with floor no. 14 followed by 27,39,50,60,69,77,84,90,95,99,100

In the worst case here total drops needed are only 14. 

For example, if the egg breaks when dropped from 14th floor then for second egg we need to test 1-13 floors to find the exact floor from which egg breaks on the drop. See table below.

Similarly, for example, egg breaks after drop from 50 floor (after testing 14,27,39), we need to test at 51-59 floors by dropping second egg to know the exact floor from where egg breaks on the drop.
 

Find Numbers For The Boxes

Find the correct number to fill in the boxes below.

These should be correct numbers!

Finding Correct Numbers For The Boxes

How boxes are placed?

Let's assume a, b, c and d are the correct numbers in the boxes.

So, 4 equations that we get are,

a + b = 8     .....(1)

c - d = 6     .....(2)

a + c = 13   .....(3)

b + d = 8    .....(4)

Subtracting (1) from (3) gives,

c - b =  5     .....(5)

Adding (2) to (4),

c + b = 14    .....(6)

Adding (5) and (6) gives,

2c = 19

c = 9.5

Putting c = 9.5 in (2),

9.5 - d = 6

Hence, d = 3.5

Putting d =3.5 in (4),

b + 3.5 = 8

b = 4.5

Putting b = 4.5 in (1),

a + 4.5 = 8

a = 3.5

So to conclude, a = 3.5, b = 4.5, c = 9.5 and d = 3.5.