Leonid Ryvkin

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Scientific publications

Preprints

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Events and Activities

Puzzles and hunts

Scavenger Hunt

Pen-and-Paper adventure

A few elementary riddles and excercises

• Subsets
  Let \(n\in \mathbb N\) be a natural number. Consider an \(n+1\)-element subset \(A\) of \(\{1,...,2n\}\). Show that \(A\) contains two distinct numbers \(a,b \), such that \(a\) divides \(b\).
  
  
• Checkerboard
  Consider a checkerboard of \(n\times n\) fields. Some fields contain pieces, others don't. Now put pieces on every field that touches (through its edges) at least two fields already containing pieces. Repeat the process until no further pieces are added. What is the minimal number of pieces on the field at the beginning of the procedure, if every field contains a piece at the end? (And why?)
  
  Hexagonal checkerboard How is the situation for a hexagonal board with sides of 6 fields, and hexagonal fields, if we need 3 neighbors of a field to already carry pieces for it to also get a piece? (communicated by Anne Marlen Prepeneit)
  
  
• Coins
  \(2n\) coins lie in a line on the table. Alice and Bob take turns in taking a coin away from the bounday of the row. Each coin is labelled with a value. In the end, after all coins are picked, the person who has taken coins with the larger value wins. In case of a draw Alice, who also takes the first turn, wins. Does someone have a winning strategy, if yes how does the strategy look?
  
  
• Burning cords
  You have two cords (or fuses), that each take exactly 60 seconds to burn. You don't know whether they burn down at a constant speed and whether they burn down in the same way. How can you use them to measure exactly 45 seconds?
  
  
• Poisoned chocolate
  Alice and Bob have an \(n\times m\) chocolate bar. The top left corner piece is poisoned. Starting with Alice, they take turns in picking a point in the chocolate bar and eating all chococolate pieces, that are right of this point and all that are below this point. They have to choose in a way that makes them eat at least one piece of chocolate. The person eating the poisoned piece loses. Does one of them have a winning strategy? If yes, who is it? Click here for an example game in which Bob loses.

  

  
  
  Poisoned chocolate (addendum) Does the answer change if the chocolate is infinitely big? (communicated by: Anne Marlen Prepeneit)
  
  
• Pirates
  A group of a thousand pirates earns 1000 pieces of gold. They have a very complicated gold distribution procedure, your task will be to find out how the gold will be distributed. First of all, all gold pieces are identically large, individual gold pieces can not be split and the pirates are totally hierarchically ordered. Pirate 1 is more important than pirate 2, pirate 2 is more important than pirate 3 etc. Their distribution procedure works as follows:
  
  

  1. The most important alive pirate proposes a distribution.
  2. The alive pirates vote whether they accept this distribution, if more than half of them are against it, the most important alive pirate is killed and they restart the procedure. Otherwise, the proposed distribution is accepted and the gold is distributed.

  
  All pirates behave perfectly rationally (other than choosing to be a pirate)m know that all other pirates also behave rationally and have the following preferences:
  
  

  1. From two options a pirate would always choose the one, where she survives.
  2. If she survives in both cases, she prefers the options where she gets more gold.
  3. If there are two cases with survival and equal amount of gold, she chooses the one where more other pirates die.

  
  So, how will the gold be distributed among the pirates and how many of them will survive?
  
  
• An unknown square
  We are interested in a square with (unknown) corners \(A,B,C,D\) and unknown side length \(a\). Assume, that for some point \(P\), we know the distance between \(A\) and \(P\), the distance between \(B\) and \(P\), and the distance between \(C\) and \(P\). Can we determine \(a\) from this data? (There is a beautiful solution using straightedge and compass).
  
  
• Prisoners' hats
  A group of 100 prisoners discovers that it will be executed (or freed) on the next day. They will have to stand in a row on a staircase, such that each prisoner sees all the prisoners in front of them but none behind them. Each prisoner will wear a hat, which is either black or white. From back to front, the henchman will ask each prisoner "What color is your hat?" If the answer is correct, then the prisoner is free, otherwise he/she is heheaded. All the remaining prisoners hear the question and answer and also hear whether the person leaves or is beheaded. Before the procedure, the prisoners have time to discuss and agree on a strategy. Assuming a reasonable level of solidarity among them, what should their strategy be and how many would be sure to survive? (communicated by: Anna Marklová)
  
  
• Toasting coordination
  A group of people are standing in a circle and want to toast (let their glasses of wine touch). It is considered impolite for the glasses to cross. Assuming that only 2 glasses are allowed to touch at a time (during one "round") and that the people don't change places and don't toast behind each others backs, how many rounds of toasting are necessary so everyone has toasted with everyone? (communicated by: Oscar Cosserat)
  
  
• Prime Sudoku
  Can you fill a Sudoku field with prime numbers, so all rows, columns and the 9 3x3-subfields contain a number at most once and sum up to prime numbers? (Additional question to which I do not know the answer: Can you fill it completely with distinct numbers). (proposed by Anna Marklová)
  
  
• Semi-integer rectangles
  Take a rectangle which is decomposed into smaller rectangles, which all have at least one integer side-length. Must the original rectangle also have an integer side length? (communicated by: Anne Marlen Prepeneit)
  
  
• Monotony
  In whatever order one puts the numbers \(1,...,n^2+1\), there will always be an \(n\)-element subsequence which is monotonous.

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