Brown math circle activities

Below is an account of math circle activities at Achievement First High School in Providence, RI. 

Fall 2021 and Spring 2022 

• November 10: The 1,2,3,4 game. Students explore what numbers you can get using 1,2,3,4 and the operations +,−,⨉,/. Can you get all (small) numbers? What's the largest number you can make? 

• November 30: Monty Hall problem. Investigation by experimenting and collecting data.

• December 14: Graph theory and the high five game. Basic problem: given n people and a number k<n, is it possible for everyone to give exactly k high fives, without giving any person a high five more than once? 

• March 1: “Sum fun” game. Place numbers in order (1,2,3,..) in one of two piles trying to avoid a+b=c in a pile. How far can you get? What if you start with 2? or start with 3? The students came up with interesting strategies to get the high score for the room. This problem is related to Ramsey theory (monochromatic arithmetic progressions). 

• March 8: Playing “Spot it” aka “Dobble”. This is a game with 55 cards, each with a number of symbols, where each pair of cards shares exactly one symbol (!). How do they do that and how many symbols are needed? The students made their own game. This relates to finite projective planes. 

• March 15: math origami, making platonic solids: tetrahedron, cube, octahedron, dodecahedron. No one managed the icosahedron!  

• March 22: towers of Hanoi. There was a friendly competition to achieve the lowest number of moves starting with different number of disks. Some students used inductive reasoning to approach the problem. 

• April 5: hexiamonds and hexiamond puzzles. Students collectively discovered all the hexiamonds. After cutting them out, they played with hexiamond packing problems. 

• April 12: Nim-like chess games. Fix a chess piece (rook, king, queen) and place it in the bottom-left of a chess board. Players alternate moving the piece toward the top-right. Player who moves the piece to the top-right wins. Is there a winning strategy? 

• April 26: Probability game. Can you guess the colors of the balls in the bag? How many guesses before you can be confident? 

• May 3: Fermi questions and orders of magnitude. How many Oreos would fit in the classroom? How long would it take to eat them? 

• May 10: Remainder game. Player 1 chooses a number 1-100. Player 2 tries to guess the number by asking its remainder after division by 2,3,..,10. How quickly can you guess? What’s the best strategy? This activity connects to the Chinese remainder theorem.  

Fall 2022 and Spring 2023

• September 19: cup stacking game. Start with some number of cups in a line. The goal is to get the cups all in one stack. A stack of K cups can jump left or right K spaces, but have to land on other cups. Problems: Come up with a way (algorithm) to get all the cups to land in your favorite spot? If the cups are numbered in increasing order, can you get them into a single stack where the numbers are also increasing? The game can also be played with a 2d grid of cups. This activity is good for inductive reasoning, experimenting with small cases, and making predictions. 

• September 26: Regular solids and beyond. Glue together triangles with k meeting at a vertex for k = 3, 4, ... What shapes do you get? Repeat with squares. 

• October 3: playing hackenbush (with one or two colored branches). 

• October 24: 1-2-3-4 game

• October 31: Probability and the birthday paradox

• November 14: the game of SET

• November 28: Intransitive dice. Students discovered/proved that the dice (2,2,4,4,9,9), (1,1,6,6,8,8), (3,3,5,5,7,7) are intransitive. Some came up with 4 dice that are intransitive. 

• December 5: making kaleidoscopes 

• February 7: Mobius magic. Cutting and pasting twisted bands and exploring the possible results. 

• February 14: sprouts game

• March 7: Hat puzzles. Everyone in a group has a hat. Players can see each others hats, but not their own.  Players come up with a strategy to guess the color of their hat without any communication. Is there a strategy that's better than randomly guessing? 

• April 4: Fermi questions

• April 25: Nim game

Fall 2023 and Spring 2024

• September 19: Pile game. Start with a pile of 10 chips. As a team you take turns dividing a pile. When you divide the pile, add the product of the numbers in the two new piles to your total score. Repeat until piles of 1 only.  Repeat the game and try to improve your high score. Play with different number of starting chips. Do small number of starting chips to understand what’s going on.

• September 26: Monty Hall.

• October 3: Playing chomp

• October 17: Towers of Hanoi

• October 31: Playing SET

• November 14: Heads or tails game. You repeatedly flip a coin. If the coin lands on heads, the pot doubles. If you flip tails, the game is over and you lose everything. You are allowed to leave the game with your winnings after a heads flip, but you must stop playing once you flip tails. What is a good strategy? 

• November 28: Fermi questions 

• December 5: the pirate game 

• February 6: estimating pi with toothpicks

• February 27: Chess nim game

• March 5: Making and playing with hexaflexagons

• March 12: Non-transitive dice

• March 19: Nim

• April 2: Fold and cut theorem