Mathematics Advent Calendar (DFG, 2013)

Goal

This is the inofficial English translation of the German Mathe-Advents-Kalender my project Matheon of the DFG from December 2013. The goal is to provide interesting and fun mathematics problems for students towards the end of high-school or beginning university education. Also people interested in mathematics are welcome to look at the problems.

December 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th,
11th, 12th, 13th, 14th, 15th, 16th, 17th, 18th, 19th, 20th,
21st, 22nd, 23rd, 24th.

Since I take part in the solving myself, I will accpet the solutions after I handed in my soltuion. Also I need time to translate the problem (literally and the context), so don't expect them to be online 12h after publication of the German original (or even earlier).

Please enjoy the riddles!

Rules

Teamwork

In the original Math (Advent) Calendar, you can only register and hand in solutions as an individually. Registration as a group is not possible due to technical as well as competition reasons. Nevertheless, we do not want and cannot suppress teamwork. Instead we explicitely encourage honest teamwork where honesty means that the focus is on mathematical exchange and learning not just exchanging the result. We encourage collaboration, because we consider it important to communicate about mathematics. The goal is to cherish the fun with mathematics and to share a joy is to double the joy.

Here the real calendar starts

December 1st: Coin flipping German

Author: John Schoenmakers

Imp Hans and imp Tim are enjoying their time with coin flipping. They trow a fair coin until the number of H (head) and T (tail) either coincides or one of them is bigger by three than the other (e.g. with the sequence Head, Head, Tail, Head, Tail, Head, Head; Hans wins).

If the numbers of H and T coincide after at least two flips, the game ends split. Otherwise Hans wins if head was three times more often than tail or Tim wins if tail is three times more often than head.

What is the probability for Tim to win for the first time after at least five games?

Hint: First, determine the probability for Tim to win dirctly the first game.
Hint2: Three times more means that three more of this kind were thrown.

  1. 0.0367
  2. 0.1256
  3. 0.3461
  4. 0.4019
  5. 0.5
  6. 0.598
  7. 0.6123
  8. 0.6667
  9. 0.7025
  10. 0.75

Decemeber 2nd: Ice soccer German

Author: Felix Günther

In the creativity shop of Santa Clause are 9 imps busy with the development of new ideas for gifts. All of them are big ice soccer fans and want to miss the top game between Borussia Kiruna and Hertha Rovaniemi by no means. Santa Clause is not convinced by the idea to send the imps off to watch the game in a group shortly before Christmas Eve; however, he knows that joint excursions can rise the motivation. Forth and back between the two options either to send many imps to the game or to let many imps work he decides for the following offer to his employees: “Tomorrow I will hood you with peaked caps which are either black and yellow – the colors of Borussia – or blue and white – the colors of Hertha. You will see the caps of all the other imps just not your own. The next moment all 9 of you have to tell me all at the same time which cap you think you are wearing. Those of you who give the right answer may watch the game.”

The same evening the imps discuss the proposal of Santa Clause. Because there is no possibility of interaction right after the hooding, they want to heck out a strategy in advance. But watching ice soccer is no fun if only few of them can join. Therefore the imps are looking for a strategy that guarantees in every case at least N imps may watch the game and N ≥ 0 should be maximized.

It means they are looking for a method that tells each imp which had he has to name (possibly dependent on the hats of the others) such that for every distribution of hats at least N imps will tell the right answer and N should be as big as possible.

Imp Pep, the old strategist, has the idea to ask Santa Clause if he may perhaps guess as the first which hat he is wearing. The imps also discuss a strategy for this case that guarantees M imps to join the game such that M is maximal.

Unsuspecting Santa Clause agrees to Peps proposal the next morning, he may in presence of all other imps say which hat he thinks to wear. Following the remaining 8 have to tell simultaneously their guesses. How many imps may now in addition go to watch ice soccer (i.e. what is M - N)?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 0

December 3rd: German

Author:


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