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Worksheet Written Homework 2

Write out solutions to the following questions neatly by hand or typed (but if you type them, make sure to use proper math typesetting). Clearly label each question. When you are proving a statement, please also state clearly what style of proof you are using (direct, contrapositive, or contradiction). Submit your solutions (as a pdf) on Canvas.

1.

A standard deck of 52 cards consists of 4 suits (hearts, diamonds, spades, and clubs) each containing 13 different values (Ace, 2, 3, …, 10, J, Q, K). If you draw some number of cards at random, you might or might not have a pair (two cards with the same value) or three cards all of the same suit. However, if you draw enough cards, you will be guaranteed to have these. For each of the following, find the smallest number of cards you would need to draw to be guaranteed having the specified cards. Prove your answers.
  1. Three of a kind (for example, three 7’s).
  2. A flush of five cards (for example, five hearts).
  3. Three cards that are either all the same suit or all different suits.

2.

The difference of sets \(A\) and \(B\text{,}\) written \(A \setminus B\text{,}\) is the set of all elements that are in \(A\) but not in \(B\text{.}\)
The empty set, written \(\emptyset\text{,}\) is the set that contains no elements.
Prove that if \(A \setminus B = A\) then \(A \cap B = \emptyset\text{.}\)

3.

Below are four mini sudoku puzzles. Each has a unique solution, but uses some new rules:
  1. A blue diagonal line contains the digits 1-4 exactly once.
  2. Digits along an arrow sum to the digit in its attached circle.
  3. Digits in cells separated by a black dot must have a ratio of 1:2. Digits in cells separated by a white dot must be consecutive (i.e., their difference is 1).
  4. Shaded squares contain even digits.
You are welcome to solve as many of these as you like, but to get credit for this question, you only need to find one number in any one of the puzzles. Then you must give a careful proof that your number is correct. Make sure you clearly state what style of proof you are using.
Four mini sudoku puzzles.
Completing multiple puzzles, or proving multiple facts about different puzzles, will earn you extra credit.
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