Update

exercises.md

@@ -211,7 +211,9 @@ Bear in mind the main purpose of this quiz is for you to test your understanding
 105. Let's have a [spiral](spiral.md) that's drawn like this: we start with a drawing hand (like e.g. that of clock) that points horizontally to the right and has length *r1*; then the hand turns around a full circle (doesn't matter in which direction), linearly increasing its length to *r2* as it goes. Find the formula for the length of this spiral (this length will be something between the circumference of a circle with radius *r1* and circumference of a circle with radius *r2*).
 106. Rounded to whole percents, what is the probability that you'll correctly answer this question?
 107. Ronald died and wasn't missed, he was just a capitalist. Every action of that bitch only served to make him rich. Things he built but always sold, patents he would always hold. As he jerked off to his brands, dick got zipped up in his pants. Ron did one last happy dance, had idiot death insurance. Do you know what kind of note this stupid's grave would be bestowed?
-108. Did you enjoy this quiz?
+108. Explain at least one of the following [chess](chess.md) concepts: fork, pin, smothered mate.
+109. There is a cube-shaped planet that has 8 houses (numbered 1 to 8), each house on one of the 8 cube vertices. Each house is inhabited by one alien (they're named *A* to *H*). Sometimes they get bored and want to switch houses with others, so they organize a big moving day in which some aliens switch houses (it's possible that everyone moves elsewhere or that just some move and some stay where they are). However they like their neighbors (aliens living in houses directly connected by the same edge), so any time this house switching occurs, at the end of the day everyone must have the same neighbors as before. How many possible ways there are to assign aliens to the houses so that they always have the same neighbors?
+110. Did you enjoy this quiz?
 
 ### Answers
 
@@ -323,7 +325,9 @@ sin(x) / cos(x) - log2(2) = tg(x) - 1*, so we get *tg(x) >= 1*. So that will hol
 105. { I hope this is right :D ~drummyfish } First imagine the graph of a polar coordinate function that says the radius of a plain circle with radius *r* depending on angle: the graph is just constant function (horizontal line) with value *r* going from 0 to *2 * pi*. Integrating this function (from 0 to 2 * pi, here we simply multiply *r* by *2 * pi* as the graph is a rectangle) will give us the formula for the circumference of circle: *2 * pi * r* -- we'll take this largely on intuition but it can be seen that this holds because we're adding constant tiny increments of length from 0 to what we know is the circle circumference (2 * pi * r). Now imagine similar function, just starting at *r1* and linearly increasing to *r2*, i.e. we just have a linear function saying the spiral radius for current angle. Again, we'll integrate this, this time getting (bottom rectangle plus upper right triangle): *2 * pi * r1 + 2 * pi * (r2 - r1) / 2*. Simplifying this we get *pi * (r1 + r2)*, which is hopefully the solution (we see this will be between the circumferences of the smaller and larger circles, also for *r1 = r2* we again get the circumference of plain circle etc.).
 106. Lol what, TBH I don't know :D The answer is probably that the question is shit because it's not even clear what it's asking, the definition of probability here is not clear (is it probability of a random "intelligent" man from the street answering it, or giving a completely randomly generated answer to it or what?). 100% might in some cases make sense (firstly we conclude that chance of guessing a number from 0 to 100 is 1/101, but then knowing this will be the answer we conclude we know it for sure, so we switch to 100% and then making further reasonings it stays stable at this value, but this probability assumes we make the reasoning we did, someone else could make a different reasoning maybe leading to other consistent answers). Haven't thought about it deeper yet though. If you know the answer let me know.
 107. Retard -- read the first letter of each sentence.
-108. yes
+108. Fork: attacking two (or more) pieces at once (often done with knight) so that the opponent can only save one. Pin: attacking a piece so that if it moves away, it will reveal another piece behind it to be taken (often pinning to king). Smothered mate: checkmate by knight in which king can't move anywhere because he's blocked by own pieces.
+109. This is counting graph [automorphisms](automorphism.md). Let's say we assign alien *X* to house 1; we can count how many possible allowed configurations there are for this case and then multiply it all by 8 (for case when *X* would be assigned to house 2, then 3, 4 etc.). Let's say neighbors of *X* are *U*, *V* and *W*. There are 3 edges going from house 1, i.e. 3 possible ways for the first neighbor, *U*, to be placed -- again, consider we put *U* in one place; we'll count the possibilities and eventually multiply them by 3. Now we have 2 edges (2 neighbor houses) remaining and 2 neighbors (*V* and *W*) to put there; again, consider one case and then multiply that by 2. Now we have *X* and all his neighbors in place, how many possible configurations are left here? There is one house that's the neighbor of both *U* and *V* and there is only one possibility of who can live there: the shared neighbor of *U* and *V* -- there is just one option so this house's inhabitant is determined. Same for *V*/*W* and *U*/*W*. That's already 7 houses assigned and the one last remaining has to be in the one house left, so in fact by placing *X* and its neighbors we've uniquely determined the rest of the houses, there's just one way. So in the end we have 8 * 3 * 2 * 1 = 48 possible ways.
+110. yes
 
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