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Following the tradition from [last year](https://www.kongregate.com/forums/7099/topics/686963) (though don't worry, I will actually deliver on FGF wins), you'll be competing on the [mathematical puzzles advent calendar from Matheon](https://www.mathekalender.de).
Scoring: you will get 1 point for giving the correct answer, and up to 6 points for your explanation, for a total of up to 7 points per problem. Here's a table with base criteria for the scoring of explanations:
ScoreMeaning

1Any minor advances towards the solution
2Genuine progress, but did not get too close to the solution
3A working solution skeleton, but has major gaps
4There is a gap in the solution, but it is not central
5A perfectly valid solution
6Demonstrates complete understanding of the problem and solution
Minor mistakes will cost 1 point at most, and will be disregarded if they pale in comparison to the rest of the solution or already cost the correct answer point.
Submit your answers and explanation for each puzzle to me by PM. You have until the end of the year to answer. I'm thinking how I can improve this so that partial results are available over time, suggestions are welcome.
Prizes: an FGF win! :D I may think on something else later. If anyone's willing to donate a prize, we'll give it out to the winner(s) too.
Leaderboard:
UserTotalDoor 1Door 2Door 3Door 4Door 5Door 6Door 7Door 8Door 9Door 10Door 11Door 12Door 13Door 14Door 15Door 16Door 17Door 18Door 19Door 20Door 21Door 22Door 23Door 24

[Funnykidsc](https://www.kongregate.com/accounts/Funnykidsc)1437767776777776777776*77
[back900](https://www.kongregate.com/accounts/back900)1347776*776*6*1*7777672773*6*77
*=nonfinal submission



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[First Door](https://www.mathekalender.de/index.php?page=problem&problemID=14)
A gift starts in one corner of a cube, then follows edges randomly 5 times. What is the probability that it ended up in the opposite corner?
Since this is the first problem, I'll say this: for a maximal score explanation, I'm expecting not only for you to answer the bonus question from the site (if the gift moved 6, 7 or 9 times), but also find the closed formula for any questions of the kind.



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[Second Door](https://www.mathekalender.de/index.php?page=problem&problemID=9)
Elves A, B and C are privately given each a different symbol, and know that these symbols belong to one of 10 sets. A the declares he knows which of the sets is the one being used, to which B replies he also knows this set, and finally to which C answers he knows the set as well. Which of these 10 sets is the one in play?



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Do we get a breakdown of the scoring for each question? Do we get told if we are wrong?



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I'll figure out the scoring soon. I'll base it on IMO standards if possible. I won't tell you if you're wrong or not before your submission is final. I will, though, be transparent with the scoring: once your submission is final, I'll tell you the score you got and why, so that if it's unfair you can appeal.
EDIT: Figured out the scoring. Now I'll figure out the deadlines. Any suggestions on both counts are appreciated.



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> *Originally posted by **[Lebossle](/forums/7099/topics/955461?page=1#11759932)**:*
> I'll figure out the scoring soon. I'll base it on IMO standards if possible. I won't tell you if you're wrong or not before your submission is final. I will, though, be transparent with the scoring: once your submission is final, I'll tell you the score you got and why, so that if it's unfair you can appeal.
>
> EDIT: Figured out the scoring. Now I'll figure out the deadlines. Any suggestions on both counts are appreciated.
Scoring posted is out of 6 not 7. What is 7th point for?



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I'm thinking actually to give immediate feedback for your first answer, but not detailed, just something like "You got the right answer, but your explanation has a big gap, so you'd get 4 points". Then the person can try to amend his answer/explanation, and that becomes his final submission for the door, or say he won't, in that case he gets the scoring I gave. After doing either, I give a more detailed explanation.
Edit because ninja'd: 6 points are for the explanation and 1 is for getting the right choice.



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[Door 3](https://www.mathekalender.de/index.php?page=problem&problemID=13)
![](https://imgur.com/y0RHZr9.png)
What is AE?
Final point if your answer is mostly geometric (rather than algebraical)



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[Door 4](https://www.mathekalender.de/index.php?page=problem&problemID=4)
Gifts are assigned to Departments according to their price: if their first digit is i, it goes to department i (i from 1 to 9). All prices doubled, but the amount of gifts assigned to each department stayed the same. Which inequality is guaranteed to be valid?
For the last point, don't forget to describe all possible valuations of (ni).



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> *Originally posted by **[Lebossle](/forums/7099/topics/955461?page=1#11763227)**:*
> For the last point, don't forget to describe all possible valuations of (ni).
Can you clarify what you mean by this? (forgive me if it's obvious)



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(ni) = (n1, n2, ..., n9), meaning a point in 9D.



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[Door 5](https://www.mathekalender.de/index.php?page=problem&problemID=7)
Given a random permutation of 22 unique values, you pick the first one after the 8th that is higher than the first 8 values. What is the probability you picked the highest value?
Last point for generalizing



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[Door 6](https://www.mathekalender.de/index.php?page=problem&problemID=18)
Mr. K sits to the right of his wife in a circular table with 10 places, and the other 4 men sit next to their wives too but all with another men to their other side. Marking the seats clockwise from 1, pairs 15, 24, 39, 68 and 710 have different genders. What is Mr. K's position?
Don't forget to complain about the poor wording choice for the last point. You can do it on [their forums](http://forum.mathekalender.de/index.php?thread/318feedback6/), but do note that they did already clarify there.



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[Door 7](https://www.mathekalender.de/index.php?page=problem&problemID=3)
If you place 1xN lines (with possibly distinct N) that do not touch, even at corners, in an 8x8 board, how many cells of the board can you fill?
Generalize for any board size for the last point.



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[Door 8](https://www.mathekalender.de/index.php?page=problem&problemID=24)
![](https://www.mathekalender.de/index.php?page=showImage&documentID=663)
Find what is 4 times the ratio between the area of the small square and the area of the big square.
For the last point, try to figure out what the "more optimal" choice being talked about by the other gnomes may be. You don't need to find any, just have a decent try at it.



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[Door 9](https://www.mathekalender.de/index.php?page=problem&problemID=1)
![](https://www.mathekalender.de/index.php?page=showImage&documentID=272)
Find (m, s) with m>0 and sqrt(m²+s²)<=m for which there is no (m', s') with m'>0 and sqrt(m'²+s'²)<=m and mm'<=40 and ss'<=800 with G(m, s) < G(m', s')



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[Door 10](https://www.mathekalender.de/index.php?page=problem&problemID=16)
![](https://www.mathekalender.de/index.php?page=showImage&documentID=721)
The above image is a flow/queueing network, where "checkpoints" are queues. Each edge has capacity of 1 unit per hour, takes the amount of hours noted in the picture to cross, and is oneway left to right. Cookie factory is the source, sending a continuous flow of 2 units per hour (starting at 8am), and sleigh runway is the sink. We are interested in the total flow that reaches the sink after 9 hours since flow start, and after 15 hours.
There are 4 flow instances/strategies you have to analyze:
* Santa's flow goes entirely through the bridge;
* Kneith's flow goes half through the mountains and half through the christmas trees;
* Rudolph's flow maximizes the total flow at both considered timestamps simultaneously;
* and elves' flow is the greedy approach, where each flow particle (elf) takes the path that will get it to the sink the soonest, even if that means waiting in queues.
For the final point, just describe how the flows end up working.
For instance the straightforward way to describe the Santa flow is "the flow into the middle path is 2 units per hour, the queue to go to the frozen lake increases by 1 unit per hour, the flow on that edge is 1 unit per hour, the flow on the bridge is 1 unit per hour starting at 10am, the flow after the bridge is 1 unit per hour starting at noon, the flow at the sink is 1 unit per hour starting at 2pm, elves starting after x time take 6h+x on their path, the total flow at 5pm is 3 units, and at 11pm is 9 units".
The Kneith flow can be described by "the flow into both top and bottom path is 1 unit per hour, there is no queueing, both flows to frozen lake and to trees is 1 unit per hour, the flow to mountains is 1 unit per hour starting at 10am, the flow from trees is 1 unit per hour starting a 11am, both flows from mountains and from bridge are 1 unit per hour starting at 2pm, the flow at the sink is 2 units per hour starting at 4pm, elves take 8 hours on their paths, the total flow at 5pm is 2 units, and at 11pm is 14 units".
(So you just have to describe Rudolph's and the elves')



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[Door 11](https://www.mathekalender.de/index.php?page=problem&problemID=6)
![](https://www.mathekalender.de/index.php?page=showImage&documentID=741)
Consider other possible arrangements of this subway network that respect all angles being multiples of 45°.
Explaining why wrong answers are true will get you the last point and help you not bomb this if you get it wrong.



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[Door 12](https://www.mathekalender.de/index.php?page=problem&problemID=22)
In a square of side 1, consider the points that are closer to the center than any of the edges. What is their area?



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[Door 13](https://www.mathekalender.de/index.php?page=problem&problemID=23)
Agents A and B have settled on a strategy where, given any set of 5 cards from a total of n, A gives B some permutation of 4 of those 5 cards, and with only that information (4 cards and their order), B is able to figure out which is the 5th card. How large can n be?
Last point includes generalizing for any amount of cards being exchanged.



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I will catch up on this, really busy sorting out Christmas things right now so sorry not been doing answers Leb.
Can deadline just be the end of the Calendar?



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> *Originally posted by **[back900](/forums/7099/topics/955461?page=1#11783690)**:*
> Can deadline just be the end of the Calendar?
According to the main post, the deadline is the end of the year. Are you asking for it to be made earlier? If so, why?



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Yeah, I wasn't sure what to put as deadline at the start, but with the low activity I'm pretty sure I'll keep it as it is.
[Door 14](https://www.mathekalender.de/index.php?page=problem&problemID=10)
Gnomes 1 to 26 sit, in order, on seats 1 to 26. The first 4 sit on random unoccupied ones, then everyone else sits on their corresponding seats, unless they are already occupied, in which case they sit on a random unoccupied one. What's the probability that gnome 26 sits on his seat?



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> *Originally posted by **[Funnykidsc](/forums/7099/topics/955461?page=1#11784156)**:*
> > *Originally posted by **[back900](/forums/7099/topics/955461?page=1#11783690)**:*
> > Can deadline just be the end of the Calendar?
>
> According to the main post, the deadline is the end of the year. Are you asking for it to be made earlier? If so, why?
>
> *Originally posted by **[Funnykidsc](/forums/7099/topics/955461?page=1#11784156)**:*
> > *Originally posted by **[back900](/forums/7099/topics/955461?page=1#11783690)**:*
> > Can deadline just be the end of the Calendar?
>
> According to the main post, the deadline is the end of the year. Are you asking for it to be made earlier? If so, why?
>
I was meaning don't change it to 7 days as then I won't get a chance to go and do them.
