Schedule

Briefly discussed the syllabus and the plans for this course.

Setup

Get a pile of sticks and two players. Also decide who will go first.

Gameplay

Alternate turns between the players. On your turn, you must pick up 1, 2, or 3 sticks from the pile.

Win Condition

Force the other player to pick up the final stick.

Write a one-to-two page reflection, answering the following questions.

1. Who are you? You can tell me anything you like. Some ideas to get you started (you do not have to answer all or just these):
   • What's your major?
   • What brought you to Sewanee?
   • Do you have any hobbies you'd like to tell me about?
   • What are your pronouns?
2. Do you have a favourite kind of puzzle or game? If so, tell me what it is. Are there any specific puzzles or games you would like to learn about in this class?
3. What do you hope to take away from this class?

We used integer division to write out a complete description of a winning strategy and the winner of every possible game of pick-up-sticks. This used the Quotient-Remainder Theorem.

We then generalized this to a variant of the game where players can pick up any number of stones from \( 1 \) to \( k \).

I asked students to think about the following variant on this game. Suppose you are allowed to pick up 1, 3, or 5 sticks on your turn. Who can win? What is the winning trategy?

We solved the 1-3-5 variant of Pick-up Sticks! This marks the conclusion of our investigations into this game.

Setup

Choose two numbers, \( n \) and \( 1 \leq k < n \). Draw \( n \) dots in a circle on your paper. Beginning at each dot, count away \( k \) dots clockwise, and draw a line connecting these the two.

Winning

You win as soon as you've finished drawing the pretty picture.

We briefly discussed the rules above. Next time we will talk about patterns arising in the pictures, and why they arise!

We discussed why some stars look different from others, and tried to find a pattern to tell us how many pieces a star will have based on its parameters (that's Mathish for the numbers \( n, k \)). We observed that the pieces were always star-like (in some cases, they are regular polygons, which we might think of as "fat stars").

In the next lecture we'll discuss exactly how to predict the total number of star pieces, and the number of points on each star!

Write a one-to-two page reflection, answering the following question.

1. How would you describe your relationship to mathematics? Do you like maths (honest answers, please)? Why or why not?

We discussed two ways of understanding the number of star "pieces" in each picture just from knowing the parameters of the star, i.e., just knowing \( n \) and \( k \). In the course of our discussion, we reviewed what factors of numbers are, and how to work with integer division.

1. The smallest number in the star piece containing \( 0 \) is the number of star pieces in the picture. In order to work this out, we learned about clock arithmetic, and applied it to solve a problem I posed at the end of the previous lecture.
2. The greatest common divisor of \( n \) and \( k \) is the number of pieces.

We discussed Euclid's Algorithm for computing the greatest common divisor. This method allows us to determine the exact number of pieces in a game of Stellartaire without ever drawing the stars themselves!

After collecting Homework 1, we discussed solutions to problems. The problems varied between the sections (I took requests in each section).

Solve each of the following problems on paper (to be turned in at the beginning of lecture).

1. You and I will play Pick-up Sticks, using the usual rules (so we can pick up 1, 2, or 3 sticks on our turn). If we play with 35 stones, would you rather go first or second? What if we play with 73 stones? What if we play with 106 stones? Explain your answers completely with reasons from class.

2. You and a friend are playing a game of variant Pick-up Sticks to decide who gets the title of GOAT. You decide together that the game will use \( 211 \) sticks, and your friend will get to go first. You really want to win this one. Would you rather play a variant where you can pick up anywhere from…

   1. \( 1 \) to \( 7 \) sticks,
   2. \( 1 \) to \( 12 \) sticks,
   3. \( 1 \) to \( 14 \) sticks, or
   4. \( 1 \) to \( 15 \) sticks?

   For each of these, give a complete explanation of who would win and why (with reasons from class).

3. Consider the variant of Pick-up Sticks where each player can pick up 1, or 4 sticks on each turn. For which numbers of sticks can each player win? What is their winning strategy?
4. Suppose an alien world has a \( 19 \) hour day and a \( 400 \) hour year. If it is \( 9 \) now, what time will it be exactly \( 12 \) years from now?
5. Draw the result of playing Stellartaire with…
   1. \( n = 12 \) and \( k = 3 \), and
   2. \( n = 12 \) and \( k = 7 \).
6. If you play a game of Stellartaire with \( n = 1634 \) and \( k = 570 \), how many pieces will the star have? Make sure to justify completely, AND show your computations by hand.

Setup

Gather two players and (optionally) a pen and paper.

Gameplay

On your turn, name a 1-digit number (anything 1 to 9). You may not name a number that has previously been named.

Winning

First player to name a triple of numbers that sum to 15 wins. If all the digits run out, the game is declared a draw.

At the end of the lecture, we saw that this game is actually Tic-Tac-Toe in disguise! The games are "isomorphs", meaning each can be transformed into the other, allowing all moves and strategies from one to be translated into moves and strategies for the other in a "reversible" way.

This is a two-player game.

Setup

Draw a big tic-tac-toe grid with a smaller tic-tac-toe grid in each of its cells.

Game Play

Each player takes turns placing their mark (X or O) in a cell on a small grid. If a player wins on a small board, they claim the corresponding square on the big grid.

Restrictions

The small grid each player plays in on their turn is decided by their opponent's previous move: they play in the grid of the corresponding cell to their opponent's last move in the smaller square. For example, if X is placed in the bottom right corner of a small grid, then the next O must be played in a cell on the bottom right corner of the big grid. These restrictions above do not apply…

1. on the first player's first turn: their is no "prior move", so they can choose to play anywhere they like.
2. if a player would be forced to play in a small grid that has already been decided, then they may choose to play anywhere they like.

Winning

The winner of the big grid is the winner of the game.

This lecture is a student-driven review for the exam. Students must bring questions to this lecture period.

Write a study guide from your notes for the first exam. All content covered before Meeting 11 is fair game for the exam, so your study guide should focus there.

Update: Exams are graded! I handed them back at the end of meeting 15.

This lecture was the first of a few discussing the name of Fractal Tic-Tac-Toe. We talked about dimension, and what it means to be one-, two-, and three-dimensional. We constructed the Koch snowflake and calculated the perimeter of each of its finite iterations: \( \mathrm{Perimeter}(S_k) = 3(\tfrac{4}{3})^k \). We observed that this means the final product has infinite perimeter.

We showed that the perimeter of the Koch snowflake is infinite. That means that it is impossible to draw the whole Koch snowflake using any physical pen!

We also began discussing the area of the Koch snowflake.

We discussed how we would try to compute the area of the Koch snowflake. This used the self-similarity idea heavily: to build \( S_{k+1} \) from \( S_k \), we add new triangles, whose areas are scaled versions of the area of the previous ones we just added…

Compute the area of \( S_1 \), the first iteration of the Koch snowflake (your class notes will help with this).

We finished a computation of the area of the Koch snowflake. We used patterns we observed in the previous lecture and a clever trick involving factoring to compute the total area as an infinite sum! The total sum turned out to be a finite number!

We discussed again the meaning of dimension. This time, we investigated dimension from the perspective of self-similarity. We did examples involving the dimension of the line, the square, and the cube. We generalized these relationships to discover \[ N = (\tfrac{1}{\epsilon})^D, \] where \( \epsilon \) is the scaling factor for the self-similar pieces, \( N \) is the number of such pieces that build up the original, and \( D \) is the dimension. Using a simple picture of the Koch snowflake, we then computed that it has dimension \( D \) where \( 4 = 3^D \), from which we concluded that \( D = \frac{\log(4)}{\log(3)} \approx 1.262 \). So the Koch snowflake has a fractional dimension!

Setup

Get two players, a hexagonal grid shaped like a rhombus, and a bunch of stones (or coins, or something else) of two different colors. Label opposing sides of the grid by the same color (so two of the sides will be black, and two will be white). Decide who will be which color.

Play

Place a stone of your color on an unoccupied hex.

Winning

A player wins when they connect the sides of the board with their color by a path of stones of their color.

You can get a printable hex board here. I suggest you play a few games with highlighters and pens. The highlighter will let you distinguish different players' moves, and the pen will let you record the order of the moves directly on the board.

We discussed strategies in the game. Students contributed ideas about how they play and who they think has a better position on the board based on…

1. whether they went first or second, and
2. what patterns they have made in their placed stones on the board.

We've been talking about fractals, but the only example we've seen has been the Koch snowflake. Do a little bit of reading about other fractals online. Write a short, one-to-two page, summary of what you read about one of these fractals—include a short explanation of how it is constructed. Discuss why you chose that one.

You can choose any of the following (or any other you might find interesting):

• Sierpinski gasket
• Cantor set
• Mandelbrot set
• Fractals in nature
• General reading (if you want more inspiration)

We played more Hex during this lecture (this time on an \( 11 \times 11 \) board). We compared the feeling and shared strategies for playing the game that we liked. Finally, we discussed some abstract properties of the game.

We proved that the second player cannot have a winning strategy. The idea was "strategy stealing": if the second player has a winning strategy, then the first player can make a random first move and then play as if they were player two. Doing so (with a minor change we discussed in lecture) gives them the win!

To justify that there can be no draws, we will (soon) use network theory!

We decided that the game board hex is played on obscures our understanding of the game.

• Seeking a simplified model, we introduced networks.
• Constructed a network that models the important aspects of Hex.
• Used this construction to write down an isomorph of Hex that is simpler to understand in some ways.

Watch this video on networks that I lovingly constructed for you. You can even access the actual notes from the actual video if you want to!

This lecture is a student-driven review session for the second exam.

Make a study guide for Exam 2 (one week hence!). This should include (at a minimum)…

1. a complete list of topics covered,
2. all important definitions,
3. explanations of what's important in those topics, and
4. as many examples as you can muster.

This is an in-class exam.

Update: Exams have been graded and returned.

This lecture was devoted to showing that there can be no ties in Hex. We used an argument with a derived graph to show this.

Logical reasoning, exemplified by some simple puzzles. In particular, we examined the problem of coloring maps.

We took a short break from map coloring to think about logic.

Codified network coloring problems and investigated some of the basic properties of network coloring.

More on network coloring problems. This time we focused on some techniques for coloring quickly, and some shortcomings of those methods.

Discussed how to find the chromatic number of a network.

We determined under what circumstances it is possible to find a walk in a network which crosses every edge exactly once.

How does the notion of logic we've been discussing in lecture match with your impressions of logic from your life? Is it intuitive to you? Write a one-to-two page reflection comparing your ideas of logic before this with the in-class notion we've been discussing.

Solve the following problems.

1. Find a proper coloring of the Petersen graph using the fewest possible number of colors. Be sure to explain why your coloring is optimal!

Discussed "naked logic", i.e., the naked single, naked pair, and naked triple techniques. The question for naked-type logic is always "what can go in [group of cells]?" See my sudoku page for practice puzzles.

Discussed the X-wing trick. See my sudoku page for practice puzzles.

Complete the sudoku puzzles handed out during the previous lecture. Try to keep track of what logic you're using as you use it.

Practiced the X-wing and discussed the Swordfish. See my sudoku page for practice puzzles.

The swordfish technique. Basically an X-wing with more rows/columns. See my sudoku page for practice puzzles.

A variant sudoku. See my sudoku page for practice puzzles.

Students should bring questions. The final is cumulative, with an emphasis on things after the midterm.

During this lecture period, I laid out what kind of questions I will ask about each topic. None of what happens on the final should be a surprise if you attended today.

Our Final Exam is scheduled for Monday, 11 December 2023 at 09:00 in Woods 121, because this section meets during Canonical Hour C.

Our Final Exam is scheduled for Saturday, 9 December 2023 at 09:00 in Woods 121, because this section meets during Canonical Hour D.