Hector's Puzzled

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]

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]

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]

You have a playlist with exactly 100 tracks (i.e., songs), numbered 1 to 100. To go to another track, there are two buttons you can press: (1) “Next,” which will take you to the next track in the list or back to song 1 if you are currently on track 100, and (2) “Random,” which will take you to a track chosen uniformly from among the 100 tracks. Pressing “Random” can restart the track you’re already listening to — this will happen 1 percent of the time you press the “Random” button. For example, if you started on track 73, and you pressed the buttons in the sequence “Random, Next, Random, Random, Next, Next, Random, Next,” you might get the following sequence of track numbers: 73, 30, 31, 67, 12, 13, 14, 89, 90. You always know the number of the track you’re currently listening to. Your goal is to get to your favorite song (on track 42, of course) with as few button presses as possible. What should your general strategy be? Assuming you start on a random track, what is the average number of button presses you would need to make to reach your favorite song? [Solution]

You are given a fair, unweighted 10-sided die with sides labeled 0 to 9 and a sheet of paper to record your score. (If the very notion of a fair 10-sided die bothers you, and you need to know what sort of three-dimensional solid it is, then forget it — you have a random number generator that gives you an integer value from 0 to 9 with equal probability. Your loss — the die was a collector’s item.) To start the game, you roll the die. Your current “score” is the number shown, divided by 10. For example, if you were to roll a 7, then your score would be 0.7. Then, you keep rolling the die over and over again. Each time you roll, if the digit shown by the die is less than or equal to the last digit of your score, then that roll becomes the new last digit of your score. Otherwise you just go ahead and roll again. The game ends when you roll a zero. For example, suppose you roll the following: 6, 2, 5, 1, 8, 1, 0. After your first roll, your score would be 0.6, After the second, it’s 0.62. You ignore the third roll, since 5 is greater than the current last digit, 2. After the fourth roll, your score is 0.621. You ignore the fifth roll, since 8 is greater than the current last digit, 1. After the sixth roll, your score is 0.6211. And after the seventh roll, the game is over — 0.6211 is your final score. What will be your average final score in this game? [Solution]