## Games and Puzzles from Bluebird Math Circle

Offer a game as a warm-up before any topic or whenever you have a few minutes for mathematical play.

Click the issue number to see the game in the context of our mathematical topic.

Classic games help students feel that they belong in mathematical communities. Bluebird Originals are new favorites that AIMC brought to the table. Enjoy!

### Card Game—Game of Set and Clans—Original—Navajo Context

 When a Navajo person introduces themselves, they name their mother’s clan (1st), their father’s clan (2nd), their maternal grandfather’s clan (3rd), and their paternal grandfather’s clan (4th). Thus every Navajo Nation member has 4 ‘attributes’ – like cards in the SET game. But unlike the game where each attribute has only 3 values, the situation here is much more complicated. Originally, there were four Navajo clans. But nowadays, there are more than 100 clans, divided into more than 20 groups. This should lead to a fascinatingly rich geometry. Would you like to invent and study this geometry? Share your thoughts with Bluebird!Image: navajowotd.com

From Issue 50

The total deck of cards for the game of SET consists of 81 cards. Each card has 4 attributes – number, shading, color, and shape. Each attribute has 3 values: the number can have the value 1, 2, or 3; the shading can have the value empty, striped, or solid; the color can have the value green, red, or purple; the shape can have the value diamond, oval, or squiggle.

We say that 3 cards form a set if for every attribute all three cards have the same value, or all three cards are of different values. If, for some attribute X, you can say, “Two cards are X but one is not X,” then these three cards do NOT form a set. You can play the game online at https://smart-games.org/en/set/start

### Guessing Game—Native American Flags—Classic—Multiple Nations

 Here are flags of eight Native American nations:

From Issue 44

Two people play a guessing game. One of them writes the name of one of these eight nations on a secret piece of paper. The other person must guess which flag the first player has written down, by asking questions. But there are rules:

1. The question must have a ‘yes’ or ‘no’ answer.
2. The question must be answerable by looking at the flag or the name of the nation… You cannot ask, “Does this nation live in Colorado?” The reason is that the second player must be able to answer the question by just looking at the flag or naming the nation whose flag it is.

Play the game a few times. The object is to guess the flag in the smallest number of questions, every time.

If the player guessing is lucky, she will get the flag on the first try. For example, she might ask: “Is it the Navajo Flag?” She might get the answer ‘yes’, and guess with just one question. But if she is not lucky, she has only eliminated one of the eight flags. How might she eliminate more flags with each question?

Remember: Do NOT assume that the player guessing is lucky. In fact, assume that he is UNlucky. In the worst case, what is the fewest number of questions with which he can win the game?

### Puzzle Games—Square to Square—Classic

From Issue 39

The starting player draws a rectangle on a grid. The sides of the rectangle lie on gridlines (so the lengths of the sides are integers). The second player then colors in the largest square possible that has a side flush with the left side of the original rectangle, so that a smaller rectangle remains. The players then alternate turns coloring in the largest square possible that touches a previous square and leaves a smaller rectangle. (All squares must have integer sides.) The game ends when the initial rectangle is filled with squares, and the person who colors in the last square wins. In the figure player 2 wins. The app at https://jrmathfestival.github.io/SquarelandArchitect/ may be useful to experiment.

### Puzzle Game—Global Math Week/Exploding Dots—Original—Navajo and Cherokee Context

 Teach us how you say the big number 175,487 in your native language.Do you say it in a base 10 way? Or base 20 way?Do you switch the order of the digits as you pronounce them?When you write the number, how do you group the digits?Here are some Navajo and Cherokee numbers to get started…

From Issue 37

Global Math Week is the annual international event where students, teachers, and parents play through 12 online puzzles or experiences called Exploding Dots, and local explorations that go with them. The mathematical theme is number systems in different bases.

### Group Word Games—Buzz and Buzzwhack—Classic

1. In the game of Buzz, players take turns counting off: 1, 2, 3…. But when a number is a multiple of 7 (that is, divisible by 7), the player must say ‘buzz’ instead.

How many times does a player say ‘buzz’ if the game goes up to 100? To 1000? (Including 100 and 1000.)
2. The game of Buzzwhack is played exactly like the game of Buzz, with the added rule that a player must say ‘whack’ if the number is a multiple of 4. If the number is a multiple of both 4 and 7, The player must make both sounds.
• How many times does a player say ‘whack’ if the game goes up to 100? To 1000? (Including 100 and 1000.)
• How many times does a player say ‘Buzzwhack’ if the game goes up to 100? To 1000? (Including 100 and 1000.)
3. The game of Buzz is a good game for exactly two players, but the game of Buzzwhack is not. Why not?

Image: Sony Computer Entertainment

### Paper-and-Pencil Game & Computer Game—The Triangle Game—Classic

From Issue 26  The Triangle Game is played on a equilateral triangle you draw. Vertices of the triangle are labeled anticlockwise by three colors which we denote by 1, 2, and 3 for convenience. Also, parallel to each side, three equally spaced lines are drawn across the triangle, thus creating 16 small triangles. The game is for two players, who take turns to label the vertices:
– a vertex on edge {1,2} may be labeled either 1 or 2, but not 3
– a vertex on edge {2,3} may be labeled either 2 or 3, but not 1
– a vertex on edge {3,1} may be labeled either 3 or 1, but not 2
– a vertex inside the big triangle may be labeled 1 or 2 or 3.

When all the vertices have been labeled, the scores of the two players are calculated as follows:
– The score of Player 1 is the number of small triangles which are labeled {1,2,3} anticlockwise.
– The score of Player 2 is the number of small triangles which are labeled {1,2,3} clockwise.
– The winner is the player with the higher score.
Try the computer-based version: https://scratch.mit.edu/projects/707742053/

### Puzzle—Equal Parts—Original—Zuni Context

 Zuni fetishes are small carvings made from stone, shells, and other materials. Within the Zuni community, the fetishes serve ceremonial purposes and depict animals and icons integral to the Zuni culture. The Zuni also create fetishes as contemporary art for museums and enthusiasts within and outside of the community.
From Issue 21  Split the figure on the grid into two equal parts (so that you can place one part on top of the other one and they completely coincide). You can move the pieces any way you want – slide, turn them around or flip them. Image: Coyote (Zuni fetish)

### Conversation Game & Computer Toy—BlueBirdBot—Original

From Issue 20  Every time the Bluebird Math Circle meets, participants listen to the bluebird song and think of math questions. Where do math questions come from? How can we pose a lot of them? Math and science people have been collecting generative questions for centuries. “Generative” has two meanings. First, the question itself isn’t too hard to ask (to generate). Second, the question starts (generates) mathematical explorations. Want to try?
Think of two nouns: Thing One and Thing Two. If you don’t know what to pick, look around you, and use the first couple of objects you notice. Put your things into these math questions:
– What shapes do you see in <Thing One>?
– How many of <Thing Two> can fit into a hogan house?
– What are similarities and differences between <Thing One> and <Thing Two>?
These questions invite us to explore size, shape, and structure. That’s where a lot of mathematics comes from! You could write a “mad libs” math question of your own, with empty slots that other people can fill in.
Visit https://naturalmath.com/BlueBirdBot/ to make various math questions. The question bot is made of templates: it needs your words to work.

Image: Jim and Tori Mullen

### Computer Game—Weave by Number—Original—Cree, Hopi, and Navajo Context

 Flat figurate numbers can be of shapes different from regular polygons, for example, rectangular numbers or trapezoidal numbers. Donna Fernandez and students from Navajo Preparatory School created a computer game about trapezoidal numbers in the languages of three First Nations – https://roshan2205.github.io/Weave-By-Number/

From Issue 14  The game is called Weave by Number. The goal of the game is to create a geometric weaving pattern found on Native American rugs. The object of the game is to stack objects to match the target number to create a rug design. To play, select a number on the rug, then use the + or – controls to change the height and width of the object. The challenge is that there may be no solution, one solution, or many solutions. If the number of objects in the stack matches the target number, then click “Submit”. If no solution can be found, then click “No solution”. If correct, the number on the grid will change color.
When you click on the numbers, they will be pronounced in various indigenous languages – Navajo, Pueblo, and Cree. This is a great way to learn an indigenous language. The game is designed with artwork depicting the landscape of the tribal lands and symbols or objects from the respective tribes.

### Puzzle—How Am I Different?—Classic—Kwantlen First Nation Context

 Four Seasons is a set of four drums made by Brandon Gabriel and Melinda Bige, Indigenous artists from the Kwantlen First Nation. The set is displayed in Surrey, British Columbia.

From Issue 8  Pick four similar but different objects, such as shapes or numbers. Think about each of the four objects and find one thing that makes that object different from the other three.

### Tabletop Game—Two-Pile Nim—Classic

From Issue 6  Nim is a quick two-player game. You’ll need some game pieces, but those could be almost anything. Stones are shown in the picture here, but you could use coins, buttons, candy, or anything else that’s convenient.
The game begins with two piles. The picture shows piles of 5 and 7 stones, but you can change the numbers in each pile when you play different games. For your turn, you can either:
– Take as many pieces from a single pile, or
– Take the same number of pieces from both piles.
The winner is the player who takes the last pieces.
Play a few games and then think about what’s happening. Are there certain positions where you’re guaranteed to win on your next move? Are there other positions where you’re guaranteed to lose on your next move? If you start with piles of 2 and 3 pieces, can you find a winning strategy? How about if you begin with piles of 5 and 7 pieces?

### Conversation Game—Piñon Nuts—Classic—Multi-Tribe Context

 The Great Basin includes the high desert regions between the Sierra Nevada and the Rocky Mountains. The Indigenous people of the region, including the Shoshone, Paiute, Washo, and Ute, traditionally gather piñon pine nuts. The tasty nuts provide excellent nutrition.

From Issue 5  You want to harvest about half of piñon nuts in a tree, and leave the rest for the forest creatures to eat and for growing new trees. How would you estimate the number of piñon nuts on a tree?
You don’t need a serious reason to estimate: just play with friends or by yourself to pass the time. How many people in a crowd, potatoes in a bag, polka dots on a shirt? Find cute and silly sets to estimate for fun. Who between your friends can estimate the closest without counting, and how do they accomplish that?

Images: Kmusser; Utah State University

### Tabletop Game—Tuknanavuhpi—Classic—Hopi Context

 Tuknanavuhpi is a two-player abstract strategy board game played by the Hopi of Arizona, United States. The game was traditionally played on a slab of stone with the board pattern etched on it.

From Issue 4  The board for the game of Tuknanavuhpi has 4 by 4 squares with their sides and diagonals intersecting in 41 points. Two players place 20 stones or grains of maize in two colors on their sides of the board. The game’s goal is to capture the other player’s piece by hopping over them. The first player who does so wins. Read more detailed rules on Wikipedia.

Image: skyruk

### Party Game—The Same Number of Friends—Classic

From Issue 3  In any group of people, at least two people have the same number of friends within the group.

Check for yourself next time you are at a party. (This also works for cousins, team members, and other types of human relations.)

Image: Spiked Math

### Puzzle—St.Ives and Oldest Stories—Classic

From Issue 2  What is the oldest story-puzzle people still retell? It’s about four thousand years old and comes from North Africa! Archeologists found it on an ancient Egyptian papyrus. It had a story-puzzle about grain, mice, and cats to practice multiplying by 7. Why 7? That’s because back then, as it is for students even now, multiplying by 7 was the hardest of the time tables x2, x3, x4… x10. Here is a more recent version from 18th century.

As I was going to St. Ives,
I met a man with seven wives,
Kits, cats, sacks, and wives,
How many were there going to St. Ives?
(Hint: you don’t really need to multiply.)

Image: vintage post card

### Conversation Game—The Scent of Time—Original—Navajo Context

 Perseverance (Ha’ahóni in Navajo) touched down in the Martian quadrant named for Arizona’s Canyon de Chelly National Monument (Tséyi’ in Navajo). The Perseverance team worked with Aaron Yazzie of NASA’s Jet Propulsion Laboratory to collaborate with Navajo Nation naming new features on Mars.

From Issue 1 Can we tell time by the smell? Ancient Chinese used scented candles to measure the passage of time. The candles had scented bands. Each hour the candle burned down to the next scent: cinnamon, anise, cloves, and so on. You could quickly smell what time it was, even at night. How can you measure the passage of time with:
Singing
Food
Walking
Animals
Weaving
Will your method work on Mars, too? What are some ways your family and community estimate time?
Image by nasa.gov