Optimal Cone

Robert’s daughter has a set of Magna-Tiles, which, as their name implies, are tiles with magnets on the edges that can be used to build various polygons and polyhedra. Some of the tiles are identical isosceles triangles with one 30 degree angle and two 75 degree angles. If you were to arrange 12 of these tiles with their 30 degree angles in the center, they would lay flat and form a regular dodecagon. If you were to put fewer (between three and 11) of those tiles together in a similar way, they would form a pyramid whose base is a regular polygon. If Robert wanted to maximize the volume contained within the resulting pyramid (presumably to store as much candy for his daughter as possible), how many tiles should he use? [Solution]

Duck Rocks

After a long night of frivolous quackery, two delirious ducks are having a difficult time finding each other in their pond. The pond happens to contain a 3×3 grid of rocks. Every minute, each duck randomly swims, independently of the other duck, from one rock to a neighboring rock in the 3×3 grid — up, down, left or right, but not diagonally. So if a duck is at the middle rock, it will next swim to one of the four side rocks with probability 1/4. From a side rock, it will swim to one of the two adjacent corner rocks or back to the middle rock, each with probability 1/3. And from a corner rock, it will swim to one of the two adjacent side rocks with probability 1/2. If the ducks both start at the middle rock, then on average, how long will it take until they’re at the same rock again? (Of course, there’s a 1/4 chance that they’ll swim in the same direction after the first minute, in which case it would only take one minute for them to be at the same rock again. But it could take much longer, if they happen to keep missing each other.) Extra credit: What if there are three or more ducks? If they all start in the middle rock, on average, how long will it take until they are all at the same rock again? [Solution]

Spelling Bee

The New York Times recently launched some new word puzzles, one of which is Spelling Bee. In this game, seven letters are arranged in a honeycomb lattice, with one letter in the center. The goal is to identify as many words that meet the following criteria: The word must be at least four letters long. The word must include the central letter. The word cannot include any letter beyond the seven given letters. Note that letters can be repeated. For example, the words GAME and AMALGAM are both acceptable words. Four-letter words are worth 1 point each, while five-letter words are worth 5 points, six-letter words are worth 6 points, seven-letter words are worth 7 points, etc. Words that use all of the seven letters in the honeycomb are known as “pangrams” and earn 7 bonus points (in addition to the points for the length of the word). Which seven-letter honeycomb results in the highest possible game score? To be a valid choice of seven letters, no letter can be repeated, it must not contain the letter S (that would be too easy) and there must be at least one pangram. [Solution]

Sock Matching

I have $N$ pairs of socks in a drawer. I pull out socks (without replacement) until I have a matching pair. On average, how many socks does it take? Extra Credit: describe the behavior of this average for large $N$. [Solution]

Consecutive Heads

If you flip a fair coin 10 times (or n times for extra credit), how likely is it that the final two flips are the first and only time you see a consecutive pair of heads? [Solution]