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One more year, one more Matheon calendar! This is [last year's](https://www.kongregate.com/forums/7099/topics//1778819), for reference. In this thread, you'll be competing on the [mathematical puzzles advent calendar from Matheon](https://www.mathekalender.de).
Scoring: no changes from last year's, 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. Suggestions are welcome as well.
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 21

[Arcanmster](https://www.kongregate.com/accounts/Arcanmster)9877777777777777
[Pigjr1](https://www.kongregate.com/accounts/Pigjr1)42777777
[BLOODYRAIN10001](https://www.kongregate.com/accounts/BLOODYRAIN10001)1974*71*
[Funnykdisc](https://www.kongregate.com/accounts/Funnykidsc)77
*=nonfinal submission
Offgame: The [Advent of Code](https://adventofcode.com/) is also going on, I made a leaderboard for Kong users with code 2062387d1afec3 , if there is enough interest we can make an FGF for it as well.



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[Door 1](https://www.mathekalender.de/index.php?page=problem&problemID=41)
Solve the following Tentai Show / SymaPix / Artist's Block:
![](https://www.mathekalender.de/index.php?page=showImage&documentID=275)



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[Door 2](https://www.mathekalender.de/index.php?page=problem&problemID=44)
You have 91 elements A, 25 elements B, and 4 elements C, you may combine two different elements to obtain the third (losing the two originals). If you end up with all of the same element, what is the maximum you may have at that point? Last point if you describe all states you can reach from any starting position.



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[Door 3](https://www.mathekalender.de/index.php?page=problem&problemID=53)
There is a big box of cookies. They are distributed as follows, and in no moment a cookie is split:
* 1 cookie is given to dog
* exactly 1/5 of the cookies are taken by A
* 1 cookie is given to dog
* exactly 1/5 of the cookies are taken by B
* 1 cookie is given to dog
* exactly 1/5 of the cookies are taken by C
* 1 cookie is given to dog
* exactly 1/5 of the cookies are taken by D
* 1 cookie is given to dog
* exactly 1/5 of the cookies are taken by E
* 1 cookie is given to dog
* exactly 1/5 of the cookies are given to each of A, B, C, D and E (that is, the cookies are divided evenly amongst the 5)
What is the least amount of cookies each person can get? And what would be other possible amounts?



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[Door 4](https://www.mathekalender.de/index.php?page=problem&problemID=50)
A soccer tournament between 4 teams ended with the following scoreboard, and with all matches having distinct scores. What were the scores?
TeamWinsLossesGoalsGoals against

Icetown2151
Frostville2135
Glacierhampton1256
Coldbury1245



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[Door 5](https://www.mathekalender.de/index.php?page=problem&problemID=62)
3 distinct numbers from 1 to 10 are distributed to elves A, B and C, where the highest number is the same as the sum of the two lowest. Then the elves have the following conversation:
* A says he can imagine 8 possibilities for B's number
* B says he can imagine 3 possibilities for C's number
* C says he knows A's number
* A says he does not know B's number
* B says he knows A's number
What could the numbers be?



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[Door 6](https://www.mathekalender.de/index.php?page=problem&problemID=47)
Find the **volume** of the sphere in the figure:
![](https://www.mathekalender.de/index.php?page=showImage&documentID=308)



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[Door 7](https://www.mathekalender.de/index.php?page=problem&problemID=56)
You start with a normal die (faces 1 to 6, opposites sum to 7), with the 1 on the top, otherwise random orientation, on the bottomleftmost square of a 101x101 chessboard. You repeatedly tilt it up or right, until you reach the toprightmost square, and sum the 201 numbers that occur as the top face of the die. What are the possible sums you may end up with?



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[Door 8](https://www.mathekalender.de/index.php?page=problem&problemID=65)
One person goes from X to Y following a random route out of the most effective ones, and so does another from Y to X, at the same moment. What's the probability that they meet halfway?
![](https://www.mathekalender.de/index.php?page=showImage&documentID=437)



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[Door 9](https://www.mathekalender.de/index.php?page=problem&problemID=68)
You have 10 bottles, an even number of them at least 2 being good. You have a machine that, given two bottles, says either that both are bad or that at least one of the two is good. You want to identify at least 2 good bottles. What is the strategy that minimizes the amount of times you use the machine on the worst case?



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[Door 10](https://www.mathekalender.de/index.php?page=problem&problemID=73)
Each one of n reindeer do the following:
 See results of all previous reindeer
 Choose p between 0 and 1
 With chance p win 1p score, with chance 1p fail to get any score
Reindeer who at the end got the best score wins.
Which p should each reindeer choose? What's the probability of each to win?



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[Door 11](https://www.mathekalender.de/index.php?page=problem&problemID=76)
A cake is split among n=100 elves as follows:
 Elf 1 gains 1/n = 1% of the cake
 Elf 2 gains 2/n = 2% of what's left
 Elf 3 gains 3% of what's left
 ...
 Elf 99 gains 99% of what's left
 Elf 100 gains all that is left
Who got the biggest piece? What if n was something else?



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[Door 12](https://www.mathekalender.de/index.php?page=problem&problemID=79)
Each of 3 elves gets a hat of color A, B or C, and can see the hats of the others. For each of the 9 possible combination of colors an elf could see, he should choose one color to guess if that combination actually happens. Find a combined strategy of the 3 elves that maximizes the amount of combinations out of hte 27 where at least one elf guesses the color of his own hat right and no elf guesses the color of his hat wrongly.
Final point: try analyzing for other amounts of elves and colors.



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[Door 13](https://www.mathekalender.de/index.php?page=problem&problemID=82)
When you tile an 8x8 square with as many 1x3 triominoes as you can, what and where can the space left be?



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[Door 14](https://www.mathekalender.de/index.php?page=problem&problemID=85)
There are 900 nonnegative integers in a hat. Their average is 1, the average of their squares is 2, and the average of their cubes is 5. What is the smallest amount of 0's that the hat can have?
Final point: What are all the combinations of numbers the hat can have?



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[Door 15](https://www.mathekalender.de/index.php?page=problem&problemID=71)
![](https://www.mathekalender.de/index.php?page=showImage&documentID=440)
There are 3 trains, all leaving Central Station at the same time, blue and yellow ones leave Reindeer Park at the same time, and all being back at Central Station by time T, where T is smallest possible. What can their schedule be?



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[Door 16](https://www.mathekalender.de/index.php?page=problem&problemID=88)
Two elves are in a line, with distance 2 between them, and with same speed of 1. How long will they take to meet each other, on average, if they start at the same time doing:
 they toss a coin do decide direction, move 1, then toss the coin again to repeat the process?
 they toss a coin do decide direction, move 1, move 1 in the other direction, then toss the coin again to repeat the process?
 they toss a coin do decide direction, move 1, move 2 in the other direction, then toss the coin again to repeat the process?



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[Door 17](https://www.mathekalender.de/index.php?page=problem&problemID=91)
You have 100 groups of 100 lightbulbs. Each individual lightbulb has probability 0.1% of being faulty. You take 3 seconds to check if an individual lightbulb is faulty. You take 3 seconds to check if there is any faulty lightbulb in a group. Many questions regarding min, max, avg, and probability of spending min or max time when checking groups or not.



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[Door 18]()
![](https://www.mathekalender.de/index.php?page=showImage&documentID=601)
Supplier Lebkuchenhausen Elisenheim Himmelsberg

Flour (g) 3500  4000 3000
Sugar (g) 2000  2000 2000
Butter (g) 3000  3000  1000
Eggs (#) 30 22 16
Hazelnut (g) 500 1000  500
Coconut (g) 0 0 400
Choose package that allows you to bake cookies with maximum total amount of hearts.



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[Door 19](https://www.mathekalender.de/index.php?page=problem&problemID=97)
The main challenge here is understanding the formulas, so I won't be explaining them here.



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[Door 20](https://www.mathekalender.de/index.php?page=problem&problemID=100)
![](https://www.mathekalender.de/index.php?page=showImage&documentID=682)
Tie random pairs of the six ends, what is the probability that you end up with exactly two loops? What about if you have 4 strings?



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[Door 21](https://www.mathekalender.de/index.php?page=problem&problemID=103)
Some simple probability, expectancy and optimal strategy calculations on second price auctions with reserve price, check the statement for all of them.



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[Door 22](https://www.mathekalender.de/index.php?page=problem&problemID=106)
There are a few red and blue baubles in a box, more blue than red. When drawing two of them, the probability that they are different colors is 1/2. When drawing 3, the probability that they are all the same color is 1/4. When drawing 3, the probability of drawing red, then blue, then red, is 0.123456. How many red and how many blue baubles are there?



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[Door 23](https://www.mathekalender.de/index.php?page=problem&problemID=109)
Can you arrange N line segments of equal length in the plane such that, for any subset of those segments, there is a halfplane that contains exactly those segments completely, the other segments having at least some portion in the other halfplane?
Also, can you arrange 4 segments in that way, such that only two of the segments overlap?
![](https://www.mathekalender.de/index.php?page=showImage&documentID=733)



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[Door 24](https://www.mathekalender.de/index.php?page=problem&problemID=59)
Which of the following shapes can be cut, along the marked lines, into two equal pieces? You can rotate and flip the pieces.
![](https://www.mathekalender.de/index.php?page=showImage&documentID=389)
