Take 13 cards of a suite (Ace, 2, 3, 4, ... , King). The game is as follows: you pick two cards, and then the third. If the third card is between the first two cards, you win. What are the chances of your winning?
If you solve the 13 card puzzle, try the same with all 52 cards
Warmup: An old sailor selling a parrot at the market bragged that his talking bird repeated every word that it heard. A young man bought the parrot and tried speaking to the bird, but it did not repeat a single word. Yes the sailor wasn't lying. How is it possible? (Answer: the parrot was deaf)
Combinatorics Review
Multistage process: Number of outcomes is the product of the number of choices for each stage
Deck of cards: Number of ways of picking a suit, and then a card within it. It is easy to see intuitively that total possibilities are same as picking any card out of the full deck
Multiple exclusive processes: Number of outcomes is the sum of the number of choices for each exclusive option
Deck of cards: Number of ways of picking a suit, and then a card within it. It is easy to see intuitively that total possibilities are same as picking any card out of the full deck
Counting the complement: Subtraction from total number of possibilities
Deck of cards: Number of ways of drawing two cards from a deck, so that two cards do not have the same value
N distinct items can be arranged in N! ways
Deck of cards: Illustrate number of ways of arranging 2-5 distinct cards
Systematic Overcounting: Division for systematic overcounting
Deck of cards: Ways of arranging 4-5 cards, where 2-3 are the same
MCDiaries - Page 124, Prob 4: Prince Ivan is on a quest to free Princess Masha, who has been imprisoned in the castle. The castle door has a simple digital lock with ten buttons, numbered 0 to 9. The door is guarded by a hungry dragon, Pashka, who likes hot dogs. The door lock can be opened by typing a secret 4-digit code, and Pashka can be distracted by hot dogs. It takes 1 second for Prince to try out a single 4-digit combination, and it takes 20 seconds for Pashka to gulp down a single hot dog. After Ivan opens the lock, it will take him one minute to fetch Masha and fly off on his magic carpet.
How many hot dogs should Ivan pack to safely rescue Princess Masha? (Answer: 503)
Suppose that Ivan knows in advance that the secret code is composed of odd digits - how many hotdogs? (Answer: 32)
Suppose that Ivan knows that secret code is composed of odd digits and has exactly one digit 5 in it - how many hotdogs? (Answer: 16)
Suppose that Ivan knows that secret code is composed of odd digits and has at least one digit 5 in it - how many hotdogs? (Answer: 19)
Introduce notion of "number of combinations" nCm
Deck of Cards: Number of ways of picking 3 cards out of a deck
Give kids an intuitive feel of why nCm works, in terms of n! arrangements of n items
Properties of nCm:
Note that choosing m items from n is the same as picking the (n-m) non-chosen items. Hence nCm = nC(n-m)
Homework Problem: Figure out your chances of winning a lucky seven game
References:
Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg
Mathematical Circle Diaries, Year 1, by Anna Burago