Update

exercises.md

@@ -61,11 +61,12 @@ This place is for suggesting programming projects that will in first place serve
 6. **[minesweeper](minesweeper.md)**: Implement the minesweeper game! It can be a purely command line program if you manage to render it well with ASCII art and make controls usable, but of course you can try making it GUI as well. There must be at least three difficulty levels that differ by board size and number of mines. First click must never land on a mine. The game must show the time it took to complete it, thumbs up for implementing a persistent top 3 score board that's saved to a file.
 7. **arbitrary size [rational numbers](rational_number.md)**: Make a library that allows working with arbitrary size rational numbers, i.e. represent each number as a pair of numerator and denominator, the number will be automatically allocating itself as much memory as it needs for storing the two internal values. Negative numbers must be supported too. It mustn't waste too much memory, i.e. whenever it changes, it will try to simplify the fraction and, if possible, decrease its size and allocate less memory. Size of the number will only be limited by amount of RAM your program can use. Furthermore implement these operations with the numbers: converting to/from the language's native numbers (with rounding if necessary), comparisons (equal, greater, greater or equal, smaller, smaller or equal), addition, subtraction, multiplication, division and printing and/or converting the number to string (at least decimal -- if the number has infinitely many fractional digits, just cut the output somewhere).
 8. **image to [ASCII art](ascii_art.md)**: Make a program that takes an RGB bitmap image and renders it with ASCII characters (i.e. prints it out to console). You can support loading the image from just one file format of your choice, possibly something simple like PPM, BMP or Farbfeld. The program must support resizing the image and it must allow to just set one dimension with keeping the aspect ratio. Thumbs up for extra features like setting brightness/contrast and so on.
-9. **educational [sorting](sorting.md) visualization**: Make a program for visualizing sorting algorithms -- it may draw real graphics (either directly to the screen or by outputting animation file) or just render ASCII art graphics, but it has to clearly show what the sorting algorithm is doing, i.e. which elements are being compared, which are swapped and if it makes good sense to highlight something else (like the pivot or already sorted part of the array), you should do it. Implement at least bubble sort, insertion sort, selection sort and quick sort. Also offer benchmark mode in which all algorithms race in sorting the same array (this can be without advanced visualization, just show e.g. number of steps for each).
-10. **3D model of [fractal](fractal.md)**: Make a program that outputs 3D model of either Siepinski triangle or Koch snowflake fractal. The output shall be some simple 3D format like obj or Collada. The model can be primitive, i.e. it can be just flat shape made of triangles which don't have to really be connected, but the program must allow specifying the number of iterations of the fractal (during invocation, e.g. as a CLI flag). Check that the model is correct by opening it in some 3D editor such as Blender.
-11. **[steganography](steganography.md)**: Make a program that hides text strings in either pictures, sounds or another text. The program must be a nice [CLI](cli.md) utility that performs both encoding and decoding -- it will allow the user to specify the string to hide (this string can be simplified to take less space, e.g. it may be converted to all caps, special characters may be removed etc.) and the data in which to embed them. The size of the string that can be encoded will of course be limited by how much space there is in the data, so you can reject or shorten the string if that's the case. The string must NOT be hidden in metadata (i.e. exif tags, file header, after the data, ...), it must be encoded in the useful data itself, i.e. in pixels of the picture, samples of the sound or characters of the text, but it mustn't be apparent that there is something hidden in the data. Use some simple technique, for example in images and sound you can often change the least significant bits without it being noticed, in text you can insert typos, hyphens, replace some periods with semicolons etc. Get creative.
-12. **[sudoku](sudoku.md) solver**: Create a program to which the user somehow passes a sudoku puzzle (in a file, through a CLI flag, interactively... the choice is yours, but passing a new puzzle mustn't require program recompilation) and the program attempts to solve it. It must first employ some basic reasoning, at very least it has to repeatedly try the elimination method, i.e. marking a set of possible values in each empty square and then reducing these sets by crossing out values that can't be in that square because the same value is in its column/row/minisquare -- wherever only one value remains in the set, it is filled in as final; this has to be repeated until no more progress is being made. If you want, you can employ other techniques as well. After this if the puzzle is still not solved, the program will resort to [brute force](brute_force.md) which has to eventually lead to solution (even if it would take too long). If the program finds that the puzzle is unsolvable, it has to report it.
-13. **language recognizer**: Make a program that will be able to learn and then recognize language of text it is given (in theory it should work for any kind of language, be it human or computer language). Specifically the program will first get *N* files, each one representing a different language (e.g. 5 books in different human languages), then it will take some other text and say to which of the initial *N* files it is linguistically most similar. For simplicity assume only plain ASCII files on input (you can just use some Unicode to ASCII utility on all input files). Use some simple [machine learning](machine_learning.md) technique such as some variant of k-NN. It will suffice if for each training example you construct a vector of some nice features, for example {average word length, vowel/consonant ratio, relative frequency of letter A, relative frequency of letter E, ...}, give each component some weight and then just find the nearest neighbour to the tested sample in this feature space (if you want to be more fancy, split the input files into parts so you get more training samples, then try k-NN with some convenient k). You shouldn't and CANNOT use neural networks, and of course you CANNOT use any machine learning library ;) You don't have to achieve great accuracy but your program should likely be able to quite reliably tell e.g. German from C++.
+9. **[pseudorandom](pseudorandom.md) tester**: Make a command line program that will test the quality of a pseudorandom generator by reading an input of bytes and performing some statistical tests on them. You have to implement at least four of the following tests: mean value of the whole sequence, ratio of 1 and 0 bits, histogram (count of each value), chi square test, correlation of consecutive values (how much a value depends on the previous one), density plot (using ASCII art) of the bytes interpreted as [X,Y] coordinates (i.e. basically a 2D histogram), monte carlo computation of some constant such as pi (with comparison to the perfect value), interpreting part of the data as rows of pixels and drawing them (with ASCII art). Of course you may implement more fancy tests. You may impose reasonable limits in your measuring (for example you don't have to report all measures absolutely precisely, for example we don't have to know exact counts of 1 bits and 0 bits, only their approximate ratio), just make the tool usable for its purpose. Test your program on data from /dev/urandom (should conclude it's random) and some non-random data, for example some txt book from Project Gutenberg.
+10. **educational [sorting](sorting.md) visualization**: Make a program for visualizing sorting algorithms -- it may draw real graphics (either directly to the screen or by outputting animation file) or just render ASCII art graphics, but it has to clearly show what the sorting algorithm is doing, i.e. which elements are being compared, which are swapped and if it makes good sense to highlight something else (like the pivot or already sorted part of the array), you should do it. Implement at least bubble sort, insertion sort, selection sort and quick sort. Also offer benchmark mode in which all algorithms race in sorting the same array (this can be without advanced visualization, just show e.g. number of steps for each).
+11. **3D model of [fractal](fractal.md)**: Make a program that outputs 3D model of either Siepinski triangle or Koch snowflake fractal. The output shall be some simple 3D format like obj or Collada. The model can be primitive, i.e. it can be just flat shape made of triangles which don't have to really be connected, but the program must allow specifying the number of iterations of the fractal (during invocation, e.g. as a CLI flag). Check that the model is correct by opening it in some 3D editor such as Blender.
+12. **[steganography](steganography.md)**: Make a program that hides text strings in either pictures, sounds or another text. The program must be a nice [CLI](cli.md) utility that performs both encoding and decoding -- it will allow the user to specify the string to hide (this string can be simplified to take less space, e.g. it may be converted to all caps, special characters may be removed etc.) and the data in which to embed them. The size of the string that can be encoded will of course be limited by how much space there is in the data, so you can reject or shorten the string if that's the case. The string must NOT be hidden in metadata (i.e. exif tags, file header, after the data, ...), it must be encoded in the useful data itself, i.e. in pixels of the picture, samples of the sound or characters of the text, but it mustn't be apparent that there is something hidden in the data. Use some simple technique, for example in images and sound you can often change the least significant bits without it being noticed, in text you can insert typos, hyphens, replace some periods with semicolons etc. Get creative.
+13. **[sudoku](sudoku.md) solver**: Create a program to which the user somehow passes a sudoku puzzle (in a file, through a CLI flag, interactively... the choice is yours, but passing a new puzzle mustn't require program recompilation) and the program attempts to solve it. It must first employ some basic reasoning, at very least it has to repeatedly try the elimination method, i.e. marking a set of possible values in each empty square and then reducing these sets by crossing out values that can't be in that square because the same value is in its column/row/minisquare -- wherever only one value remains in the set, it is filled in as final; this has to be repeated until no more progress is being made. If you want, you can employ other techniques as well. After this if the puzzle is still not solved, the program will resort to [brute force](brute_force.md) which has to eventually lead to solution (even if it would take too long). If the program finds that the puzzle is unsolvable, it has to report it.
+14. **language recognizer**: Make a program that will be able to learn and then recognize language of text it is given (in theory it should work for any kind of language, be it human or computer language). Specifically the program will first get *N* files, each one representing a different language (e.g. 5 books in different human languages), then it will take some other text and say to which of the initial *N* files it is linguistically most similar. For simplicity assume only plain ASCII files on input (you can just use some Unicode to ASCII utility on all input files). Use some simple [machine learning](machine_learning.md) technique such as some variant of k-NN. It will suffice if for each training example you construct a vector of some nice features, for example {average word length, vowel/consonant ratio, relative frequency of letter A, relative frequency of letter E, ...}, give each component some weight and then just find the nearest neighbour to the tested sample in this feature space (if you want to be more fancy, split the input files into parts so you get more training samples, then try k-NN with some convenient k). You shouldn't and CANNOT use neural networks, and of course you CANNOT use any machine learning library ;) You don't have to achieve great accuracy but your program should likely be able to quite reliably tell e.g. German from C++.
 
 ### Level 3: Hard, *Ultra Violence*
 
@@ -192,7 +193,9 @@ Bear in mind the main purpose of this quiz is for you to test your understanding
 89. We know that an [ellipse](ellipse.md) is a set of points in 2D plane that have constant sum of [distances](distance.md) to some two given points that are called *focal points*. What if we instead consider a taxicab distance (computed as distance alongside X axis plus distance alongside Y axis)? Consider the constant sum of distances to always be set higher than the taxicab distance of the two focal points. What shape will we get? Just describe the shape and intuitively show why it looks like that.
 90. What does [Turing tarpit](turing_tarpit.md) mean?
 91. =fgtnmtg qlcowj jakju lm vglcnr gjv dm gocl gjv qk vcjU
-92. Did you enjoy this quiz?
+92. Please solve the following inequality: *sin(2 * x) / (2 - 2 * sin^2(x)) - log2(1 / 8^(-1/3)) >= 0*, mathematically write exactly which values of *x* will satisfy it. Don't use calculator, ok? But you can look up goniometric formulas etc.
+93. Given continuous differentiable function *f(x)*, derive the formula for computing the length of the curve of the function graph on interval *[x1,x2]*. No need to provide 100% formal proofs, you can use intuition as long as you get the correct formula and show it works on a few examples. For example the length of the graph of function *f(x) = x* on interval *[0,1]* will be *sqrt(2)* (holds from Pythagorean theorem). Compute the length of curve of the graph of *f(x) = sin(x)* on interval *[0,2 * pi]*.
+94. Did you enjoy this quiz?
 
 ### Answers
 
@@ -287,7 +290,10 @@ Bear in mind the main purpose of this quiz is for you to test your understanding
 89. Kind of octagon but with unevenly long sides; a rectangle with bevelled corners, i.e. two horizontal sides, two vertical sides, two 45 degree walls and two 135 degree walls. We can imagine taxicab distance from given point like sort of a diamond, it creates 4 quadrants around the point, in each the distance increases linearly in diagonal direction -- regions of constant distance here form 45 degree angled squares. Boundaries between these quadrants form a cross of infinite size. Taking two different points these two crosses will overlap and form 9 regions (draw it): top-left, top-middle, top-right, middle-left etc. Examining each of the regions we will find that it either keeps the increasing direction the same (if both overlaid directions are the same) or that some principal direction cancels out and leave the sum increasing only in one principal direction -- basically we find that in each of those regions the sum increases linearly in one of 8 directions separated by 45 degrees (except for the middle region where the sum is constant). It's also clear the heightmap has to stay continuous as both of the summed functions are continuous. From all this we can deduce the shape basically.
 90. It's a [Turing complete](turing_complete.md) system (typically a [programming language](programming_language.md)) that's however extremely hard to use for any practical programming, i.e. it can be seen as a programming language in which it is theoretically possible to program anything (anything programmable in any other language) but practically it's impossible to program anything significant because of the complicated nature of that language. This terms is related to [esoteric languages](esolang.md).
 91. [Earth](earth.md) or jvpcG. The cipher reverses the ASCII string and xors every byte (that's not a space) with 0x02 (i.e. flips the second lowest bit) -- don't bitch too much about this being too arbitrary, you can notice the string is reversed by the last character being uppercase and the first one being special char (?), then you can kind of recognize the words as the encoded chars are close to their decoded versions and the lengths of the words also hint on the words (for example a question is quite likely to start with "What").
-92. yes
+92. Let's simplify the left-hand side: *sin(2 * x) / (2 - 2 * sin^2(x)) - log2(1 / 8^(-1/3)) = 2 * sin(x) * cos(x) / (2 * (1 - sin^2(x))) - log2(8^1/3) = 2 * sin(x) * cos(x) / (2 * cos^2(x)) - log2(2) =
+sin(x) / cos(x) - log2(2) = tg(x) - 1*, so we get *tg(x) >= 1*. So that will hold when *pi/4 + pi * n <= x < pi/2 + pi * n*, *n* is an integer.
+93. Considering an infinitely small non-zero interval *dx*, and the graph height increase over this interval *dy*, the distance increase (from Pythagorean theorem) on this interval will be *sqrt(dx^2 + dy^2)*. We can replace *dy* by *tan(alpha) * dx*. By definition tangent of the function's angle at a certain point is its derivative, so we can also replace *tan(alpha)* by derivative of the function, *f'(x)*. So we get length increase *sqrt(dx^2 + f'(x)^2 * dx^2) = sqrt(dx^2 * (1 + f'(x)^2)) = dx * sqrt(1 + f'(x)^2)*. Now to add infinitely many values over infinitely small intervals we use integrals, so to add all these small length increases we can write the final formula: *length(x1,x2) = Integral(x1,x2) sqrt(1 + f'(x)^2) dx*. Testing this on *f(x) = x* from 0 to 1 we get the expected *length(0,1) = Integral(0,1) sqrt(1 + 1^2) dx = sqrt(2)*. For *f(x) = sin(x)* from 0 to *2 * pi* we get *length(0,2 * pi) = Integral(0,2 * pi) sqrt(1 + cos^2(x)) dx ~= 7.64*, which seems about right (it's a bit more than 2 * pi).
+94. yes
 
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