More Games on Graphs

Description

• How will this game change if we play it on a different graph?
• What if we changed the rules or invented a different, but similar game?

The Games

• Calmly at at table
  • The Mail Carrier's Tour
  • The Painter of Lines in the Street
  • The Colorful Houses of Tourist Town
  • The Colorful Sidewalks of Tourist Town
• Outdoors or in the gym
  • Toss 'n' Sort
  • White Cells and Intruders
  • Quibble, Crouch, Cornerstone, Quoink
  • Squelch and Pop

Instructions

1. Invite students to play games, invent new ones, and to discuss what they learn as they do so.

2. Play the games outdoors or in a gym so the students can move about the graphs, as well as with markers that can be moved on graphs drawn on paper.

3. Using the stories about Gertrude, Superperson and the Monster and the games presented here as models, encourage students to invent other stories with other characters and situations that they can use to present and act out their games.

4. Talk about what is alike and what is different about these games. Use these ideas to invent other games.

5. Connect the workings of the games to the properties of graphs and draw connections between the game boards and the idea of a graph as a mathematical object.

Games that are well suited for playing indoors at a table with graphs drawn on paper

• Mail Carrier's Tour

  Imagine that the graph is the map of a neighborhood. The edges are streets and at every vertex is a house. The mail carrier doesn't want to walk down any more streets than necessary to pick up and deliver mail to all of the houses. What is the best route to take? Where should they put a drop box where the truck will leave all the mail that needs to be delivered in the neighborhood?

  ( Here's an example.)

  This problem seems very much like The Painter of Lines in the Street. Try both games on the same graph and see what you think.

  ---

• The Painter of Lines in the Street

  The person who paints the lines in the streets works hard and doesn't want to have to spend time and energy lugging painting equipment when it is not necessary. What is the best route to take as you paint the streets so that all the streets get painted, and the only time that you are on a street is when you are painting it? Can you end up where you started?

  ( Here's an example. )

  This problem seems very much like The Mail Carrier's Tour. Try both problems on the same graph and see what you think.

• ---

  The Colorful Houses of Tourist Town

  In Tourist Town there is something interesting to see in every house, and the people who live there want the tourists to visit them all. The City Council has decided to buy enough paint to repaint all of the houses. If tourists leave a house that they have just visited and look down a street and see a house that is the same color, they might think that it is just like the house that they just left and not visit it. So the City Council wants to repaint the houses so that no house is the same color as the next house down the street in any direction. How many different colors of paint will the City Council need to buy?

  ( Here's an example.)

  This problem is related to the one of coloring the sidewalks of Tourist Town, but they are not quite the same, as you will see if you do them on the same graphs. These in turn are related to map coloring.

  ---

• The Colorful Sidewalks of Tourist Town

  Since all the grownups got to paint their houses, the City Council decided that the children should be able to paint the sidewalks in bright and interesting colors. They don't want the tourists to ever get bored by walking down two streets in a row whose sidewalks are the same color. How many different colors of sidewalk paint must the City Council buy? Experiment by drawing graphs and coloring their "sidewalks". ( Here's an example. )

  This problem is related to the one of coloring the houses of Tourist Town, but they are not quite the same, as you will see if you do them on the same graphs. These in turn are related to map coloring. See also, Quibble, Crouch, Cornerstone, Quoink

Active games for playing outdoors or in the gym.

Huge game boards can be drawn with chalk on a parking lot or painted with lawn paint (the same non-toxic paint used to mark lines in playing fields) on the grass. Both have their drawbacks: the lines scuff off, skinned knees on concrete, the paint is expensive. Recently we have tried using brightly colored plastic tape available inexpensively from buiding supply stores. It can be fastened to tufts of grass with masking tape, or fastened to the ground with heavy wire bent into the shape of a "U". (Be very careful to pick up all the wire when you are finished.)

Toss 'n' Sort

• Description

  This game illustrates routing and deadlock in networks. Each student is like a "packet" of information or a message which has a destination. In order to get to that destination, the traveler can only travel along a route when it is clear. Since the object is for all the travelers to get home, everyone has to work together to find a way for this to happen.

• Materials

  • Graph on the playing area.
  • One ball that is easy to throw and catch
  • Optional: Extra balls make the game easier

• Number of players

  When there is one less player than the number of vertices in the graph, the game is the most challenging. Decreasing the number of players or increasing the number of balls makes it easier. (Why?)

• Set-up

  • Players will stand in vertices of the graph and throw and catch balls. So the graph should not be so large that the players cannot throw all the way across.
  • Label each vertex of the graph uniquely---with letter, numbers, names of places, symbols, etc.
  • Prepare a set of cards that match the labels for the vertices. Make the cards so that players can wear them---pinned to their clothing, or on strings around their necks, for example.

• How to play

  • Players take their places at the vertices of the graph. At least one vertex will be left empty.
  • Distribute the cards randomly among the players.
  • Give the ball to one of the players
  • The object of the game is for all of the players to get to the vertex whose label matches the card that they have.
  • Players can run down the edge to the next vertex only if:
    • the vertex is empty
    • they are holding the ball
  • A player who has the ball can throw it to any other player.

• Variations

  • Depending on the size of the graph, the game can be quite difficult when only one vertex is left empty and just one player at a time can move. Decreasing the number of players to leave two or more vertices empty will make the game easier. Increasing the number of balls permits more than one player to move at a time. In some cases this makes the game easier, and in others it increases the level of confusion.

  • Have a player stand in every vertex. Use one or more pairs of matching balls (i.e., two red, two blue, etc.). Any two players who are holding identical balls can exchange places.

  • Play the game or any of its variations without talking, signaling, or using any other means to tell other players what to do. In addition to being strangely quiet, it requires that all participants have some kind of strategy for working the game out. Sometimes it helps to have a planning session before beginning the period of silence.

  • Use a stopwatch to time how long it takes to play a round, then try to beat previous records for shortest time.

  • Count how many times the ball is tossed from start to finish, and try to beat previous records for fewest tosses.

  • Invent a rhythm of foot-stamping and hand-clapping that is 4 to 8 beats long. When the rhythm is being tapped out, the player holding the ball throws it to someone else. When the rhythm is complete, the player who is holding the ball must run to a neighboring vertex.

  • Invent a rap or rhyme which all players chant to accompany the rhythm (above). This has the same effect as a rule which prohibits telling other players what to do, but is considerably more fun.

  • Label the vertices of the graph so that the labels can only be seen when the player at the vertex. Have players conceal the cards which tell the label of their "home" vertex from other players. Whenever a player arrives in the vertex that matches the card that she is carrying, she crouches down to signal to the other players that she is in her home vertex. A player who is crouched in her home vertex can still catch the ball and travel to neighboring vertices at any time.

White Cells and Intruders

• Description

  Like white blood cells which surround and destroy viruses, bacteria, and other foreign substances in our bodies, players in this game try to find a strategy to surround and capture whoever is "it".

• Materials

  • A graph drawn on the playing area
  • A bright colored flag or scarf for identifying the Intruder(s)

• Number of players

  • 1 person who is the Intruder (or more as a variation)
  • Two or more players who are white blood cells. (The more white blood cells there are, the easier it is to capture the intruder.)

• Set-up

  Draw the graph on the playing area

• How to play

  • Each of the White Blood Cells chooses a vertex of the graph and stands in or near it.
  • The Intruder, who wears or carries a bright colored scarf or flag, chooses a vertex and stands in or near it.
  • The object of the game is for the White Blood Cells to completely surround the Intruder.
  • The White Blood Cells and the Intruder take turns moving. The White Blood cells go first.
  • Players can move as follows:
    • White Blood Cells: When it is the White Blood Cells' turn, one White Blood Cell can run along one edge to the next .
    • Intruder: When it is the Intruder's turn, she can run along any number of edges to a different vertex, but if she passes through a vertex that is occupied by a White Blood Cell she is captured.

• Description

  This game is based on coloring problems. The classic coloring problem is Map Coloring and asks the question: If I have a map of some area that is divided into regions, how many colors will I need to color those regions so that no two neighboring regions are the same color? In the mathematical coloring of maps, just as in the coloring of actual political maps, two regions are considered neighbors when they meet along a line. Thus the U.S. states of Colorado and Arizona can be colored the same color.

  To play, one player stands in each region of the map and can assume one of several different body positions. When no player has the same body position as any of her neighbors, the game is finished.

• Materials

  A map drawn in the playing area so that the regions are large enough for a person to stand in. (A graph will work, you can just ignore the vertices. It is somewhat more versatile because it can be used for other games and variations of this one.)

• Number of players

  As many players as there are regions in the map

• Set-up

  Draw the map in the playing area.

• How to play

  • The complete list of body positions:
    • Crouch: player is squatting
    • Quibble: player stands on the left or right foot, switching back and forth so as not to get tired.
    • Cornerstone: player stands stiff, straight, and tall with hands at sides
    • Quoink: player jumps up and down rapidly in place. Even the most energetic of players will eventually get tired, so this is the most undersirable position and should be avoided whenever possible.
    • Confused: sitting crosslegged. This is the starting position for all players in the game, and also what a player does when she doesn't know what other position she ought to take.
  • Play begins with each player sitting cross-legged in one region of the map. The cross-legged sitting position is called "Confused".
  • The object of the game is for each player to take a position (either Crouch, Quibble, Cornerstone, or Quoink) that is different from all of her neighbors.
  • Players experiment with different positions until they find a combination that works. (Note: It is impossible to construct a map that would require more than 4 colors (or body positions) to color it. Many maps require only two or three.)

• Variations

  (Note: for some variations of the game, more positions that the four described above will need to be invented by the players.)
  • Coloring vertices: Each player positions herself in a vertex of a graph. A player in vertex node cannot have the same position as a player in another vertex that is connected to it by a edge. (See also, The Colorful Houses of Tourist Town

  • Coloring edges: Each player positions herself on an edge of the graph. The game is finished when all of the players on edges that touch a single vertex have different positions. See also, The Colorful Houses of Tourist Town.

  • Position partners: After the players placed themselves on the map, assign some or all of them to groups which must, in the finished product, have the same position. Note that this must be done with a certain amount of care, because if two of these players are neighbors, the game cannot finish.

  • Position puzzle: Assign all the players to positions (Quibble, Crouch, etc.) ahead of time. Then they must place themselves on the map or graph so that no two players with the same position are in adjacent places.

Description

• Number of Players

  As many people as there are vertices in the graph.

• Materials

  • Graph drawn on the playing area.
  • Some kind of distinguishing clothing (hat, scarf, flag, shirt, etc.) to distinguish the Squelchers from the Poppers.

• Set-up

  • Draw the graph in the playing area
  • Have the students divide themselves into Squelchers and Poppers
  • You might want to experiment to see how many Squelchers and how many Poppers will make a good game for the graph you have chosen to play on. Or you may want to leave that up to the students to decide as they play.

• How to Play

  • The object of the game for the Squelchers is to squelch the Poppers. The object for the Poppers is to not get squelched.
  • Each of the Poppers chooses a (different) vertex of the graph to stand in. Then the Squelchers position themselves, one to a vertex.
  • A Popper who is standing up is popped. A player who is popped must stay at a vertex unless she is running to pop a neighboring team member who has been squelched.
  • Two Poppers may only occupy a single vertex when one of them is popping the other.
  • A Popper is squelched when a Squelcher runs down an edge to the vertex where she is standing and taps her on the head. A squelched Popper must crouch down.
  • A Popper who has been squelched can be popped when an unsquelched Popper runs down an edge and taps her on top of the head. When a squelched Popper is popped, she pops up from her crouch and stands tall. A Popper who has been squelched cannot be popped if a Squelcher is standing in the vertex with her.
  • Squelchers can prevent any Peeper from being popped by standing in the vertex with her.
  • A squelched Popper cannot leave her vertex to pop another Popper.
  • Squelchers may pass squelched poppers. No other players may pass one another.
  • The game ends either when all of the Poppers have been squelched or the Squelchers give up.

What next