Taking Students Out for a Ride: Using a Board Game to
Teach Graph Theory
Darren Lim
Siena College
515 Loudon Road
Loudonville, New York
[email protected]
ABSTRACT
sites across the internet. In a class devoted towards ObjectOriented Design and Advanced Java programming, the topic
of graph theory was presented in a novel fashion: by using
a board game. Games have been source material for computer sciences teachers for decades. Monopoly [15] and even
Chutes and Ladders [1] have been examples for computer
simulation and modeling. C-jump [2] can be used to introduce students to fundamental constructs of programming
such as conditionals, increments, loops, and jumps. Chess
[12, 8], Checkers [13] and Connect Four [4] have all been used
in discussions regarding artificial intelligence. The dice game
Pig has been used to illustrate concepts in a CS1 course [11].
Even video games have been used as class material in CS
curriculums [9, 17].
The board game Ticket to Ride [3] was used as an introduction to graph theory in an sophomore-level class called
“Object-Oriented Design and Programming.” Its gameplay
elements require the understanding of both basic graph theory topics such as vertices, edges and paths, and algorithmic concepts such as finding the longest path in a graph.
After discussing how to play the game during class, the corresponding homework assignment required the students to
implement a graph data structure and a searching method
for their structures.
The organization of this paper is as follows. The next section outlines the basic rules and gameplay of Ticket to Ride.
The following sections discuss how the game can be used
as a tool to present concepts of graph theory to students.
Finally, a concluding section will discuss future use of the
game in larger scale projects.
This paper describes the use of a board game as a device for
introducing graph theory to computer science students. By
experiencing a hands-on demonstration of graphs, students
can better understand the basic principles of graph theory
and can better design algorithms and programs which manipulate graph data. The programming assignment tied to
the game forces the students to come to grips with the algorithmic aspects of graph theory, including the proper choice
of data structures and the implementation of graph algorithms.
Categories and Subject Descriptors
K.3.2 [Computer and Information Science Education]:
Computer Science Education—Curriculum; G.2.2 [Discrete
Mathematics]: Graph Theory
General Terms
None
Keywords
Computer Science Education, Graph Theory, Games
1. INTRODUCTION
Graphs and graph theory are fundamental aspects of a
computer science curriculum. Graphs appear in many different courses, including data structures, networks, algorithms,
and formal languages. However, the topic of graph theory
usually falls under the auspices of a mathematics department. In this situation, graph theory is formally introduced
using strict mathematical definitions and notations, accompanied with problem solving via proofs.
Typical resources for introducing graph theory to students
include texts [6, 7, 16], software [14, 5, 10], and countless
2.
TICKET TO RIDE: THE GAME
The gameboard itself is a graphical model of nineteenth
century North America, complete with vertices representing cities and edges representing “train routes” from one
city to another. The board is shown in Figure 1. The
object of Ticket to Ride is to accumulate the most points
among all players. Each player is awarded points during the
game when he or she claims a unclaimed route on the board.
These routes are claimed by discarding colored cards from
a hand that match, in color and quantity, a particular route
on the board. That player marks the route as claimed by
themselves by placing train tokens at the route’s location
on the board. At the end of the game, points are tallied up
for each route claimed. At this time, additional points are
awarded (or deducted) based on special Destination Tickets,
which are acquired throughout the game. After these spe-
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367
Figure 1: The Ticket to Ride Board Game.
cial award points are distributed, the player with the most
points is declared the winner.
2.1 A Player’s Turn
During each player’s turn, he or she may choose to do one
of the following three maneuvers:
• Draw train car card(s)
• Claim a route
• Draw Destination Tickets
2.1.1
Drawing train car cards
Before a player can claim a route between cities, he or she
must first build up a hand of train cards. In the game, there
are eight colored train cards, each of which depicts a different type of train car (Box, Caboose, Coal, Freight, Hopper,
Passenger, Reefer, and Tanker). A special locomotive acts
as a wildcard. They are all shown in Figure 2. Each player
starts the game with four random train car cards in their
hand. At each turn, five cards are displayed face up for
choosing from the deck. The player can either choose a face
up card, or choose a card from the top of the deck. If the
player does not choose a face up wildcard (a card which can
represent any color), the player may choose a second card:
either a non-wildcard face up card or a card from the top
of the train card deck. As a player chooses a face up card,
a replacement is put in its place from the top of the deck.
If at any time, three locomotive cards are face up, the five
face up cards are discarded, and five new face up cards are
taken from the top of the deck.
2.1.2
Figure 2: Train Cards. These are drawn and subsequently played from a player’s hand in order to
claim a route.
player has claimed three routes: Portland ⇔ Salt Lake City,
Helena ⇔ Salt Lake City, and Salt Lake City ⇔ Denver.
A route has one of nine distinct colors, eight of which
correspond to the colors of the train cards. A grey colored
route, can be claimed by any set of colored cards (or wildcards). For example, the grey route between Houston and
New Orleans of length two could be claimed by discarding
two orange train cards, two green train cards, one green and
one wildcard, etc.
2.1.3
Claiming routes
Drawing Destination Tickets
At the beginning of the game, each player draws three destination tickets and keeps at least two of them. An example
is shown in Figure 3.
A Destination Ticket is a card with two highlighted cities
and a point value. The point value represents the amount of
points the player holding the challenge card would receive
at the end of the game if that player controls a path of
routes between both cities on the card. During regular play,
One individual route can be claimed during a player’s
turn. Each route has a distinct color and length. To claim
an orange route of length three, for example, a player must
discard exactly three orange (or wildcard) cards from his
or her hand. Players designate a claimed route by placing
train tokens of their color on the board. Once a route is
claimed by any player, it can not be reclaimed by any other
player for the remainder of the game. In Figure 4, the green
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4, students can see that Helena is incident to seven routes,
while Salt Lake City is incident to merely five. Thus, the
degree of these two vertices are seven and five, respectively.
Figure 3: A Destination Ticket. Completing a path
between Miami and Los Angeles garners twenty
points
a player may choose to draw three challenge cards; he or she
may then discard up to two of those cards drawn.
2.2 Determining the winner
Every player starts with forty-five train tokens, with which
to claim train routes. Each player gets one final turn whenever any one player has zero, one or two train tokens available to play. At the end of the game, players are rewarded
points for each route claimed, according to Table 1.
Route Length
1
2
3
4
5
6
Figure 4: Claimed Routes in the Northwest
3.2
Properties of Edges
After examining the board, students can easily see that
the seventy-eight edges are not all the same; each edge has
one of nine colors and a length between one and six. This
has two implications. First, students will have to properly
model their edges as a class entity, instead of a primitive int
datatype. This gives the students another chance to practice
their Java class designing skills.
The second implication is that the two edge properties allow the instructor to introduce several topics of algorithmic
graph theory. Because each edge has a specific color, the
topic of edge coloring could be discussed. Since each edge
has a specific length, that value can be used as a weight
for discussions on minimum spanning tree algorithms, and
shortest path algorithms.
Points Awarded
1
2
4
6
10
15
Table 1: Route Point Values
Players are also awarded points for each successfully met
destination ticket that they possess. A Destination Ticket
is met only if a path exists between both cities designated
on the ticket. A player can connect cities with a path using
only edges claimed by himself or herself. If such a path does
not exist, the points are deducted from the player for that
Destination Ticket. Finally, ten bonus points are awarded
for the player with the longest continuous path of captured
routes.
3.3
3. TICKET TO RIDE: CONCEPTS IN GRAPH
THEORY
One of the first things that students notice about the
routes on the gameboard is that some are doubled between
two adjacent cities. Several vertices, such as Seattle and
Portland in Figure 4 are connected by two edges instead of
one. Most of the board is modeled as a simple graph; however, the presence of multiple edges illustrates that the full
graph is an example of a multigraph. This fact will have immediate implications on students’ choices of data structures.
3.4
By showing students the gameboard graph (for the remainder of the paper, the graph of the full gameboard will
be referred to as the full graph), many simple concepts can
be easily discussed.
Simple Graphs and Multigraphs
Subgraphs and Components
Students quickly realize that the board has more edges
than any one player could claim. Thus, the set of vertices(cities) and edges(routes) covered by a player’s train
token set is a smaller graph, namely a player’s Edge-Induced
subgraph. The subgraph clearly need not include the entire
set of vertices nor edges. Subgraphs need not even be fully
connected; two separate subgraphs can exist on the board.
This allows the instructor to describe separate components
of graphs.
3.1 Properties of Vertices
Each vertex of the full graph represents one of thirtysix different cities from nineteenth century North America.
Students can learn what exactly is meant by vertices and
edges, rather than digesting the classic, formal definition of
a graph (a pair G = (V, E)...). They can see a real example
of information that is more easily described and presented
using graphs. By using train tokens, the concepts of adjacency incidence, and degree can be quickly shown. In Figure
3.5
Paths and Cycles
Determining whether a Destination Ticket has been met is
done by finding a path between two cities (vertices). Paths
369
card.txt
Input Files
edge.txt
Sault St. Marie : Nashville : 8
Little Rock : Oklahoma City
Sault St. Marie : Oklahoma City : 9
Little Rock : Saint Louis
Nashville : Saint Louis
Oklahoma City : Kansas City
Omaha : Kansas City
Omaha : Duluth
Winnipeg : Duluth
Winnipeg : Helena
Sault St. Marie : Winnipeg
Sault St. Marie : Duluth
Sault St. Marie : Toronto
Sault St. Marie : Montreal
Toronto : Montreal
Toronto : Pittsburgh
Toronto : Duluth
Figure 5: A Cycle in the South
Table 2: Route Point Values
in the graph can be easily described using the board and
train tokens. In Figure 4, a path of claimed routes exists
between Portland and Denver (through Salt Lake City). An
instructor can use tokens to illustrate the subtle differences
between a path, a trail, and a walk.
The concept of cycles can also be illustrated by claiming a
route (or a path of edges) to two cities already connected by
the player such as in Figure 5. Students soon recognize that,
from a strategic standpoint, creating cycles in their collection of claimed train routes does not increase their coverage
of cities and therefore does not help them meet their Destination Tickets.
a final score of 94) using the Java programming language.
As a guide, a sample jarfile application and sample input
data were provided.
To accomplish this (besides the usual File I/O processing
and string parsing), the students need to understand several
concepts from algorithmic graph theory.
4.1
Graph Data Representation
Implementing graphs is a classic topic of a Data Structures course. Typically, students need to understand the
two methods for representing graph data: adjacency matrices and adjacency lists. The board of Ticket to Ride makes
their decision an interesting one, since the full adjacency
matrix is quite sparse. Although not required for the assignment, students implementing the full graph must also think
about how to properly handle multiple edges between nodes.
In some cases, these multiple edges aren’t alike; sometimes
these edges have different colors.
After describing both data structures in class, most students initially implemented their graphs using adjacency
matrices. This is not surprising, since two-dimensional arrays are a more familiar data structure for them. Students
quickly realized that there is a trade-off in their choice of
data structure. Adjacency matrices allow for easier calculations of edge scores, while adjacency lists allow for simpler
implementations of Destination Ticket scoring.
3.6 Trees and Spanning Trees
Since a player doesn’t benefit from creating cycles in his
or her subgraph of claimed routes (except in the case where
claiming a route blocks other players from joining or extending paths), the ideal selection of edges would form a forest
or tree, depending on the number of components. A nearperfect strategy for a player is to create a spanning tree of
the graph. By definition, a spanning tree of the full graph
would guarantee that any Destination Ticket would be fulfilled.
Players do not have enough train tokens to claim a spanning tree of the full graph. Thus, to fulfill the Destination
cards in their hands, the best strategy is to capture a minimal spanning tree or forest of a subset of vertices (including
those on their Destination Tickets) of the full graph.
4.2
4. TICKET TO RIDE: CONCEPTS IN COMPUTER SCIENCE
Depth-First, or Breadth-First, Search
The scoring aspect of the individual edges is a simple
lookup in the full graph data structure. The Destination
Tickets require a more sophisticated algorithm for determining whether the player should have points awarded or deducted for each Destination Ticket in his or her hand. This
is best implemented by using a Depth-First Search (DFS)
algorithm.
In class, the DFS algorithm was shown as a method for
determining the number of components in a graph. It was
shown in a non-recursive form, such that an adjustment in
the “next vertex to visit” array would drastically change the
visiting order. Treating the array as a stack would produce
a Depth-First Search; treating the array as a queue would
produce a Breadth-First Search.
The programming assignment for the class was to compute
the final score of a player, based on the following input.
• A list of routes claimed by the player.
• A list of Destination Tickets possessed by the player.
The input files for their programs were two files: card.txt,
which contained the Destination Ticket Information (two
cities and a value); and edge.txt, which contained a list of
routes claim. Sample input is shown in Table 2.
Students had two weeks to write a program which would
calculate a player’s final score (the example from Table 2 has
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4.3 Path Algorithms
Given a Destination Ticket, such as the one in Figure 3, a
player may want to know the shortest path between cities,
in order to make the route-claiming process as efficient as
possible. This information is obtained through Dijkstra’s
algorithm. Implementing Dijkstra’s algorithm is a classic
topic for algorithms classes, since a careful choice of data
structures (adjacency lists and binary heaps) can affect the
big-O complexity.
A final ten point bonus is awarded to the player with the
longest path on the board. Finding the longest path in an
arbitrary graph is NP-Complete. Explaining to students
why finding the longest path in the general case is such a
difficult problem allows the instructor to talk about several
other difficult graph problems, such as Hamiltonian Cycle,
Vertex Coloring, Independent Set, and Graph Isomorphism.
5. CONCLUSION AND FUTURE WORK
Using Ticket to Ride can be an excellent way to introduce graph theory to computer science students. It allows
the instructor to visually demonstrate both simple and complex concepts. The game itself offers many different ideas
for problems and programming assignments. Although a
hands-on approach is certainly no replacement for proofs
and mathematical formalism, using this boardgame can supplement the graph theory educational experience.
In the future, the assignment can be extended to a halfsemester project, where teams would have to maintain player’s
claimed routes and card hands throughout the game. Their
projects would update scores on each turn and would also
identify the player who current has the longest continuous
path. A full-semester project for Ticket to Ride would be
the creation of a computerized player, replete with in-game
positional analysis, route-claiming decision making based on
the computer’s hand, and tactical heuristics with regards to
preventing other players from reaching their target paths.
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