In an effort to make travel more efficient, the queen has announced that all tunnels will be made one-way. If the direction of travel for each tunnel is chosen independently and at random, what is the probability that the bees can still get from any chamber to any other chamber?

]]>Can you establish an upper bound on the number of numbers in the jar?

]]>Unfortunately, the software is a little finicky. It only responds to queries of one kind: given three players $A$, $B$, and $C$, it can report whether the statement “$A$ has a lower batting average than $B$, who has a lower batting average than $C$” is true.

If Kanga and Roo want to create a ranking of $N$ of their friends, determine the least number of queries they must make.

]]>The game proceeds as follows. Each turn, one of the bears takes $2k$ drops of honey from one of the pots, eats $k$ of the drops, and places the other $k$ drops in the other pot. The bears alternate until no valid move is available, at which point the last bear to move wins.

For which values of $a$ and $b$ does the bear that goes first have a winning strategy?

]]>To avoid things getting messy on the ark, he had all of the animals order themselves in a single queue. He required the $i^{\textrm{th}}$ pair of animals to place themselves in the queue such that they were separated by exactly $i$ other animals.

Here is an example of such an arrangement, known as a Langford pairing:

Let $n$ be the total number of pairs of animals. The given example has $n=4$. Determine all values of $n$ for which a Langford pairing exists.

]]>As always, there’s a catch. The crafty djinn marks $n$ of the cookies from $1$ to $n$ using some icing. He then arranges them in some fixed order $\pi([n])$ hidden away from Albert’s view.

Now, if Albert can name a subset $S$ of $[n]$ such that some permutation of $S$ is a contiguous subsequence of $\pi([n])$, he gets to eat $|S|$ cookies from the jar.

All queries (Albert’s choices for $S$) must be written down before any of them are processed. What is the least number of queries Albert can make to guarantee he will receive at least one cookie?

$\textbf{Bonus:}$ Determine, for fixed $n$ and $q$, the set of $q$ queries that maximizes the expected number of cookies Albert will get to eat.

]]>Now, well distributed orders (like $(10, 10, 10)$) are typically more satisfactory than unbalanced orders (such as $(29, 1, 0)$). After all, we don’t want everyone fighting over the lone spring roll! Let the quality $Q$ of a given order be modeled by the function $Q(x,y,z) = xyz$.

Anand knows how much his family will eat, so he wants $x+y+z = 30$. He will select a tuple $(x,y,z)$ at random from all those that satisfy this constraint. Determine the expected value of $Q(x,y,z)$, the quality of the order.

]]>Billy wants feedback on small, medium, and large sizes. Taking out his knife, he cuts the brownie into $5$ pieces, and rearranges them to form $3$ smaller squares of distinct sizes. Find a way this can be done.

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