Puzzles and hunts
Scavenger Hunt
Alfonso
Garmendia and I have created a scavenger hunt for the participants of the SFARS seminar, that we organize.
It is still available
here, the access code to start the hunt is
"Request Mission".
A few elementary riddles and excercises
I know some of the following from Benjamin Böhme, who in turn credits Wim Martens. If you know a source I should
refer to for some of them, please let me know.
- 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)
Click here or scroll down for more riddles.