
Note: This lesson is the "Toss 'n' Sort" activity which is freely
available from the
"More
Games on Graphs" page of the
MegaMath
web site.
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.
(Here are diagrams of
some
interesting sample graphs.)
 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?)
Setup

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 uniquelywith 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 thempinned 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 footstamping and handclapping 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.
Skills:
 Problem solving / reasoning / communication / connections
 Strategy development
 Cooperative problem solving
