Easy | General |

Light bulbs are numbered 1 to 100, and kept off initially. First person comes and toggles all the bulbs which are multiple of 1, i.e. he switches all bulbs to on. Second person toggles all multiples of 2, i.e he turns of even bulbs. Third person comes and toggles all multiples of 3. This process continues till 100 persons pass. After this, how many bulbs are ON?

Hint

Consider the bulb number 9.

Answer

10 bulbs

Solution

We notice that for a perfect square (like 9), the number of factors are always odd, for example:

Number of factors of (16) = # [1,2,4,8,16] = 5

Note that for non-square numbers, factors are even.

As a factor toggles the state of a bulb, bulb number 9 will be toggled by 1,3 & 9. Thus bulb number 9 will switch ON, OFF, ON respectively. Note that odd number of factors cause bulb 9 to be ON at the end.

We note that for odd number of factors is the cause of bulb staying on at the end. Similarly every squared digit bulb will be switched on, and rest will remain off after all factors toggle. Thus the bulbs 1,4,9....81,100 are ON, at the end. Hence 10 bulbs are on.

Number of factors of (16) = # [1,2,4,8,16] = 5

Note that for non-square numbers, factors are even.

As a factor toggles the state of a bulb, bulb number 9 will be toggled by 1,3 & 9. Thus bulb number 9 will switch ON, OFF, ON respectively. Note that odd number of factors cause bulb 9 to be ON at the end.

We note that for odd number of factors is the cause of bulb staying on at the end. Similarly every squared digit bulb will be switched on, and rest will remain off after all factors toggle. Thus the bulbs 1,4,9....81,100 are ON, at the end. Hence 10 bulbs are on.

Easy | Probability |

We have a weighted coin which shows a Head with probability p, (0.5<p<1). How do we get a fair toss from this? That is, how do we toss this coin in such a way that we can have probability of winning = loosing = 50%?

Hint

Clearly we cannot have a fair toss in a single flip of this coin. But by tossing this coin 2 times, we can assign the outputs to "win" or "loose", such that we have a 50% chance for both.

Answer

Toss 2 times, mapping HT to "win", TH to "loose" and repeat the process otherwise.

Solution

Toss the coin twice. If consecutive Heads-Tails appears (HT), we "win". In case of (TH), we "loose". If (TT) or (HH) appears, repeat the process. Probability of infinite repetition (p*p*.....) + (1-p)*(1-p)*.... = 0 + 0 = 0;

Probability of H-T and T-H is equal hence it's fair now.

I proposed a faster method, "lets keep tossing the coin to form a sequence of H's & T's . I win if HT appears before TH" . Was I bluffing?

Actually yes, the probability of HT before TH is just p in second game.

Probability of H-T and T-H is equal hence it's fair now.

I proposed a faster method, "lets keep tossing the coin to form a sequence of H's & T's . I win if HT appears before TH" . Was I bluffing?

Actually yes, the probability of HT before TH is just p in second game.

Source: Common

Enable Like and Comment Easy | Probability |

Spiderman has two girlfriends, Mary Jane & Gwen Stacy. After every mission, he rushes to the central subway. Since spidey is a nice man, much impartial, he takes which-ever train arrives first. From subway, one series go towards Mary's place, and another series move towards Stacy. Trains from either series appear every 10 minutes. Also, Peter Parker sticks with the train which arrives first.

But somehow, he notices that he is spending 9 times more dates with Mary Jane than Stacy. Can you explain why?

But somehow, he notices that he is spending 9 times more dates with Mary Jane than Stacy. Can you explain why?

Hint

Do you think ratio should be 50:50? Do you think the gap between each train's arrival is 5 minutes? Can a train arrive always earlier than the other and still be at every 10 minutes gap?

Solution

Train to Mary Jane's place comes at say times 0 and 10, while the train to Stacy's place comes at times 1 and 11. So 9/10 times, train to Mary Jane's place is appearing earlier.

Easy | Strategy |

An Egg breaks only if dropped from above a threshold floor, within this 100 story building. Every time you drop the egg, it is counted an attempt. You are given two eggs, find an algorithm to deduce the threshold floor, with minimum number of attempts in worst case!

Hint

If we had only 1 egg, we would go linearly from 1 to 100. Having an extra egg gives an opportunity to jump (skip some floors from testing). When 1st egg breaks, second egg moves linearly. Think why first egg should not move in constant jumps, but rather decreasing jumps! This will give 14 steps in the worst case.

Answer

It can be done in 14 steps in the worst case.

Solution

A solution for minimum steps in worst case is the smallest integer greater than or equal to the positive solution of n(n+1)/2=100...which gives 14....

start at 14th floor, if egg breaks start linearly from 1, if it does not break drop the egg from 14+13 = 27th floor, and so on....

My friend was asked only one puzzle in his interview, "3 eggs". He got the job.

start at 14th floor, if egg breaks start linearly from 1, if it does not break drop the egg from 14+13 = 27th floor, and so on....

My friend was asked only one puzzle in his interview, "3 eggs". He got the job.

Source: Quant Interview

Enable Like and Comment Easy | General |

Without explicitly calculating, find out which is bigger: e^Pi or Pi^e?

Answer

e^Pi

Solution

This is a common placement test question. Since e^x = 1 + x…., we have e^x>1+x. Now, conveneintly choosing x = (Pi / e -1) and solving, we get e^Pi > Pi^e.

Another way is to observe for f(x) = x^(1/x), differentiating: f'(x) = f(x)*(1/x^2)*(1- ln x); f is decreasing for x > e so pi^(1/pi) < e^(1/e); hence e^(pi)> pi^(e)

Another approach is to assume e^Pi > Pi^e <=> Pi*lne > e*ln Pi <=> lne/e > lnPi/Pi. Now, notice that f(x) = lnx/x is decreasing function since f'(x) is negative in range (e<x<pi), hence above assumption is true!

Another way is to observe for f(x) = x^(1/x), differentiating: f'(x) = f(x)*(1/x^2)*(1- ln x); f is decreasing for x > e so pi^(1/pi) < e^(1/e); hence e^(pi)> pi^(e)

Another approach is to assume e^Pi > Pi^e <=> Pi*lne > e*ln Pi <=> lne/e > lnPi/Pi. Now, notice that f(x) = lnx/x is decreasing function since f'(x) is negative in range (e<x<pi), hence above assumption is true!

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