Update

exercises.md

@@ -125,11 +125,12 @@ Here are some questions to test your LRS related knowledge :D
 24. In 3D computer [graphics](graphics.md) what's the difference between [shading](shading.md) and drawing [shadows](shadow.md)?
 25. Can we say that the traditional feed forward [neural networks](neural_network.md) are [Turing complete](turing_complete.md)? Explain why or why not.
 26. Wicw mx uum yvfe bbt uhmtf ok?
-27. Does the statement "10 does not equal 10" logically [imply](implication.md) that intelligent alien life exists?
-28. What is the principle of [asymmetric cryptography](asymmetric_cryptography.md) and why is it called *asymmetric*?
-29. What is the main reason for [Earth](earth.md) having seasons (summer, winter, ...)?
-30. WARNING: VERY HARD. There are two integers, both greater than 1 and smaller than 100. *P* knows their product, *S* knows their sum. They have this conversation: *P* says: I don't know the numbers. *S* says: I know you don't, I don't know them either. *P* says: now I know them. *S* says: now I know them too. What are the numbers? To solve this you are allowed to use a programming language, pen and paper etc. { Holy shit this took me like a whole day. ~drummyfish }
-31. Did you enjoy this quiz?
+27. What is the *Big O* time [complexity](complexity.md) of worst case scenario for [binary search](binary_search.md)?
+28. Does the statement "10 does not equal 10" logically [imply](implication.md) that intelligent alien life exists?
+29. What is the principle of [asymmetric cryptography](asymmetric_cryptography.md) and why is it called *asymmetric*?
+30. What is the main reason for [Earth](earth.md) having seasons (summer, winter, ...)?
+31. WARNING: VERY HARD. There are two integers, both greater than 1 and smaller than 100. *P* knows their product, *S* knows their sum. They have this conversation: *P* says: I don't know the numbers. *S* says: I know you don't, I don't know them either. *P* says: now I know them. *S* says: now I know them too. What are the numbers? To solve this you are allowed to use a programming language, pen and paper etc. { Holy shit this took me like a whole day. ~drummyfish }
+32. Did you enjoy this quiz?
 
 ### Answers
 
@@ -159,11 +160,12 @@ Here are some questions to test your LRS related knowledge :D
 24. Shading is the process of computing surface color of 3D objects, typically depending on the object's material and done by GPU programs called [shaders](shader.md); shading involves for example applying textures, normal mapping and mainly lighting -- though it can make pixels lighter and darker, e.g. depending on surface normal, it only applies local models of light, i.e. doesn't compute true shadows cast by other objects. On the other hand computing shadows uses some method that works with the scene as a whole to compute true shadowing of objects by other objects.
 25. We can't really talk about Turing completeness of plain neural networks, they cannot be Turing complete because they just transform fixed length input into fixed length output -- a Turing complete model of computation must be able to operate with arbitrarily large input and output. In theory we can replace any neural network with logic circuit or even just plain lookup table. Significance of neural networks doesn't lie in their computational power but rather in their efficiency, i.e. a relatively small and simple neural network may replace what would otherwise be an enormously large and complicated circuit.
 26. two (or txq); The cipher offsets each letter by its position.
-27. Yes, a false statement implies anything.
-28. The main difference against symmetric cryptography is we have two keys instead of one, one (private) for encrypting and one (public) for decrypting -- neither key can be used for the other task. Therefore encryption and decryption processes differ greatly (while in symmetric cryptography it's essentially the same, using the same key, just in reversed way), the problem looks different in one direction that the other, hence it is called *asymmetric*.
-29. It's not the distance from the Sun (the distance doesn't change that much and it also wouldn't explain why opposite hemispheres have opposite seasons) but the tilted Earth axis -- the tilt changes the maximum height to which the Sun rises above any specific spot and so the angle under which it shines on the that spot; the [cosine](cos.md) of this angle says how much energy the place gets from the Sun (similarly to how we use cosine to determine how much light is reflected off of a surface in [shaders](shader.md)).
-30. 4 and 13, solution: make a table, columns are first integer, rows are second (remember, both *P* and *S* can be making their own table like this too). Cross out whole bottom triangle (symmetric values). *P* doesn't know the numbers, so cross out all combinations of two primes (he would know such numbers as they have only a unique product). *S* knew *P* didn't know the numbers, so the sum also mustn't be a sum of two primes (if the sum could be written as a prime plus prime, *S* couldn't have known that *P* didn't know the numbers, the numbers may have been those two primes and *P* would have known them). This means you can cross out all such numbers -- these are all bottom-left-to-top-right diagonals that go through at least one already crossed out number (combination of primes), as these diagonal have constant sum. Now *P* has a table like this with relatively few numbers left -- if he now leaves in only the numbers that make the product he knows, he'll very likely be left with only one combination of numbers -- there are still many combinations like this, but only the situation when the numbers are set to be 4 and 13 allows *S* to also deduce the numbers after *P* declares he knows the numbers -- this is because *S* knows the combination lies on one specific constant-sum diagonal and 4-13 lie on the only diagonal that in this situation has a unique product within the reduced table. So with some other combinations *P* could deduce the numbers too, but only with 4-13 *S* can finally say he knows them too.
-31. yes
+27. *log2(n)*; Binary search works by splitting the data in half, then moving inside the half which contains the searched item, recursively splitting that one in half again and so on -- for this the algorithm will perform at worst as many steps as how many times we can divide the data in halves which is what base 2 logarithm tells us.
+28. Yes, a false statement implies anything.
+29. The main difference against symmetric cryptography is we have two keys instead of one, one (private) for encrypting and one (public) for decrypting -- neither key can be used for the other task. Therefore encryption and decryption processes differ greatly (while in symmetric cryptography it's essentially the same, using the same key, just in reversed way), the problem looks different in one direction that the other, hence it is called *asymmetric*.
+30. It's not the distance from the Sun (the distance doesn't change that much and it also wouldn't explain why opposite hemispheres have opposite seasons) but the tilted Earth axis -- the tilt changes the maximum height to which the Sun rises above any specific spot and so the angle under which it shines on the that spot; the [cosine](cos.md) of this angle says how much energy the place gets from the Sun (similarly to how we use cosine to determine how much light is reflected off of a surface in [shaders](shader.md)).
+31. 4 and 13, solution: make a table, columns are first integer, rows are second (remember, both *P* and *S* can be making their own table like this too). Cross out whole bottom triangle (symmetric values). *P* doesn't know the numbers, so cross out all combinations of two primes (he would know such numbers as they have only a unique product). *S* knew *P* didn't know the numbers, so the sum also mustn't be a sum of two primes (if the sum could be written as a prime plus prime, *S* couldn't have known that *P* didn't know the numbers, the numbers may have been those two primes and *P* would have known them). This means you can cross out all such numbers -- these are all bottom-left-to-top-right diagonals that go through at least one already crossed out number (combination of primes), as these diagonal have constant sum. Now *P* has a table like this with relatively few numbers left -- if he now leaves in only the numbers that make the product he knows, he'll very likely be left with only one combination of numbers -- there are still many combinations like this, but only the situation when the numbers are set to be 4 and 13 allows *S* to also deduce the numbers after *P* declares he knows the numbers -- this is because *S* knows the combination lies on one specific constant-sum diagonal and 4-13 lie on the only diagonal that in this situation has a unique product within the reduced table. So with some other combinations *P* could deduce the numbers too, but only with 4-13 *S* can finally say he knows them too.
+32. yes
 
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