Math and logic puzzle

Question 1

you have to bucket a 5/4 bucket and a 3/4 bucket, weirdly shaped, halh of the height is not half of the volume. An infinite amount of water. How do you measure 4/4 of water?

Answer 1

You should fill the 5/4 put in into the 3/4. Now you have 3/4 and 2/4. You throw the 3/4 away put the 2/4 in it and fill the 5/4 with that you fill the 3/4 or better the 1/4 missing and you are done.

Question 2

Flatten pills: you have 20 bottles of pills. 19 have 1 gram pills one has 1.1g pills. how do you identify the heavy one by measuring with a precise scale only once??

Answer 2

You have one try. So most of the bottle needs to be used on the scale. If you do not use more than 1 you can not identify it. Also, you have to use them differently otherwise how do you identify the heavy one in the buck? So… Take a different number of pills from everyone 1 for first 2 from second etc… then you will be able to identify it,

Question 3

Mutiledted dominos, can you cover a 8x8 checkboard with mutileted corners with 31 dominos (that covers 2 places?)?

Answer 3

No even if you can cover 31x2 62 squares and you have 62 squares a domino always cover one black and one white. if you cut 2 blacks you are screwed

Question 4

combination n objects take k

Answer 4

you create permutations of length k of n objects but now order does not matter so you divide by k! the rearrangement of the k-subset objects n!/(k!(n-k)!)

Question 5

Blue eye islander: on an island all the blue eyed people have to leave. There is on flight per night. You can see other people eye but you do not know your own and they can not tell you. Do not know how many blue eyes but know there is at least one

Answer 5

If only one person you can see all other no blue eyes. So you leave. If 2 you are not sure, cause you see another one with blue eyes. but if she does not leave the first night it means you also have blue eyes and on the second night you both leave. So it takes c+1 with c is the number of people with bljue eye

Question 6

Permutations: how many string of n characters

Answer 6

n!

Question 7

Relation between log_10 P and log_2P

Answer 7

Remember P=10^log_10(P)= 2^log_2(P) log_10(P) = log_2(P)/log_2(10) So they only differ by a constant factor.

Question 8

9 balls one is heavier. Identify it using only 2 uses of a scale that tells you if left or right is heavier.

Answer 8

weight 3 and 3. Then either one of those 3 to identify which one is heavier or if they are the same the remaining 2 you have not weighted the first time.

Question 9

sum of power of 2 say 2^0+2^1…2^n

Answer 9

• you can see it using binary try to sum up them in binary format and you will see it. it is 2^(n+1)-1

Question 10

What is the sum ffor 1…N integers?

Answer 10

you can see this by writing them and grouping in couple them so that they sum up to n+1 (N+1)*N/2

Question 11

Permutations with length

Answer 11

n characters you want k character string n!/(n-k)! because you can choose n characters for the fisr n-1 for the second etc but you stop at n-k+1

Question 12

Given a generator of unbiased bernoulli numbers (0 or 1 with p=0.5), create a biased bernoulli trial generator (generate 0 or 1 with the specified 0 < p < 1)

Question 13

How to write a function to make a biased coin from a fair coin and vice versa

Question 14

How do we handle categorical variables in decision trees?

Question 15

Random forest:

What happens when we have correlated features in our data?

Question 16

Random Forest

What if instead of finding the best split, we randomly select a few splits and just select the best from them. Will it work?

Question 17

RAndom forest

Is it easy to parallelize training of a random forest model? How can we do it?

Question 18

What is gradient boosting trees?

Question 19

What’s the difference between random forest and gradient boosting?

Question 20

What are the main parameters in the gradient boosting model?

Question 21

How do you select the number of trees in the gradient boosting model?

Question 22

Feature importance in gradient boosting trees  —  what are possible options?

Question 23

Is it possible to parallelize training of a gradient boosting model? How to do it? ‍⭐️

Question 24

What is Adam? What’s the main difference between Adam and SGD?

Question 25

Do we want to have a constant learning rate or we better change it throughout training?

Question 26

Why do we actually need convolutions? Can’t we use fully-connected layers for that?

Question 27

Are CNNs resistant to rotations? What happens to the predictions of a CNN if an image is rotated?

Question 28

ow does max pooling work? Are there other pooling techniques?

Question 29

What kind of CNN architectures for classification do you know?

Question 30

What is object detection? Do you know any architectures for that?

Question 31

hat is object segmentation? Do you know any architectures for that?

Question 32

What is bag of words? How we can use it for text classification?

Question 33

What are N-grams? How can we use them?

Question 34

How large should be N for our bag of words when using N-grams?

Question 35

What is TF-IDF? How is it useful for text classification?

Question 36

Which model would you use for text classification with bag of words features?

Question 37

Would you prefer gradient boosting trees model or logistic regression when doing text classification with bag of words?

Question 39

What are word embeddings? Why are they useful? Do you know Word2Vec?

Question 40

If you have a sentence with multiple words, you may need to combine multiple word embeddings into one. How would you do it?

Question 41

How can we use CNN for text classification?

Question 42

What is the ranking problem? Which models can you use to solve them?

Question 43

What are good unsupervised baselines for text information retrieval?

Question 44

Can we formulate the search problem as a classification problem? How? ‍

Question 45

What are good baselines when building a recommender system?

Question 46

What is collaborative filtering?

Question 47

How we can incorporate implicit feedback (clicks, etc) into our recommender systems? ‍⭐️

Question 48

What is the cold start problem?

Question 49

Possible approaches to solving the cold start problem?

Question 50

How would you select a representative sample of search queries from 5 million queries?

not algorithmically more on bird -ey view of the features you need

Answer 50

Some key features need to be kept in mind while selecting a representative sample.

Diversity: A sample must be as diverse as the 5 million search queries. It should be sensitive to all the local differences between the search query and should keep those features in mind.

Consistency: We need to make sure that any change we see in our sample data is also reflected in the true population which is the 5 million queries.

Transparency: It is extremely important to decide the appropriate sample size and structure so that it is a true representative. These properties of a sample should be discussed to ensure that the results are accurate.

Question 51

There are 6 marbles in a bag, 1 is white. You reach in the bag 100 times. After drawing a marble, it is placed back in the bag. What is the probability of drawing the white marble at least once?

Answer 51

The probability of drawing out at least one marble is the complement of probability of drawing not a single white marble at all. Therefore, we’ll calculate the Probability of drawing all non-white marbles over a hundred times and subtract by 1:

P(White at least once) = 1 – [P(Non-white marbles) ^ 100] = 1 - [(5/6) ^ 100]

Question 52

You call 3 random friends who live in Seattle and ask each independently if it’s raining. Each of your friends has a 2/3 chance of telling you the truth and a 1/3 chance of lying. All three say “yes”. What’s the probability it’s actually raining?

Answer 52

We have to find the probability of raining in Seattle given that all three friends said ‘Yes’.

Therefore, we are trying to find: P(rain | yes, yes, yes)

Using Bayes Theorem, our equation will now be:

P(rain | yes, yes, yes) =

P(yes,yes,yes|rain)*P(rain) / [P(yes,yes,yes|rain)*P(not rain) + P(yes,yes,yes|rain)*P(not rain)]

We have the following values:

P(yes, yes, yes | rain) = 2/3 ^ 3 = 8/27

P(yes, yes, yes | not rain) = 1/3 ^ 3 = 1/27

P(rain) = R (it is not given in question, so we’ll assume R)

P(not rain) = 1 - R

Substituting these values in equation we get:

P(rain | yes, yes, yes) = 8P/(7P + 1)

Question 53

You have 2 dice. What is the probability of getting at least one 4? Also find out the probability of getting at least one 4 if you have n dice.

Answer 53

For 2 die the probability of getting at least one four is: P(at least 1 four) = 1 – P(No four) = 1 – 5/65/6 = 1 – (5/6)^2 = 11/36

Following the pattern above, the probability with n dice will be: P(at least 1 four) = 1 – P(No four) = 1 – 5/6n