Teaching objectives
Can you draw a figure without lifting your pencil and without retracing any line? This question, which sounds like a game, hides a mathematical secret almost 300 years old. This lab is designed for primary-school students and takes them from tracing a simple house all the way to solving the famous Königsberg bridges problem that Leonhard Euler unravelled in 1736.
What you'll learn
Students start by tracing figures with clicks — they click one vertex, then another, and the "pencil" travels along the edge — and gradually discover why some figures can be drawn in a single stroke and others cannot.
- Students learn to identify vertices and edges of a graph, and to count the degree of each vertex (how many edges meet at it).
- Each vertex is classified as even or odd, and Euler's rule is applied: a graph can be traversed in a single stroke if it has 0 or exactly 2 vertices of odd degree.
- Students practise the strategy of starting at an odd vertex to find the correct path.
- In the final rooms students work with real bridges (Königsberg) and learn how to move a bridge so that a full traversal becomes possible or even circular (returning to the starting point).
- Room 12 introduces the rule through an emergent revelation that students verify using data they have collected themselves.
Key mathematical ideas
- A drawing of vertices and lines is a graph; the lines are edges and the points are vertices.
- The degree of a vertex is the number of edges that leave it.
- Parity as a key mathematical concept.
- Euler's Theorem (1736): a connected graph has an Eulerian path (traversing every edge exactly once) if and only if it has 0 or 2 vertices of odd degree. With 0 odd vertices the path can be closed (returning to the start).
- A graph with 4 or more odd vertices does not admit any Eulerian path: this is exactly what Euler proved with the 7 bridges of Königsberg.
- Modifying an edge (moving a bridge) changes the degrees of two vertices and can make the graph Eulerian.
Room-by-room contents
Room 1 · Welcome
The challenge is introduced: draw a small house without lifting the pencil and without retracing any line. It opens the 300-year mathematical mystery that runs through the whole lab.
Student tasks
- Read the challenge and look at the house.
- Think about whether it is possible before moving on.
Room 2 · Your first trace
First interactive figure: a simple triangle. The student clicks vertices in order to trace all edges without repeating any.
Student tasks
- Click a vertex to start.
- Keep clicking adjacent vertices until the trace is complete.
Room 3 · The house without a door
A house without the door line: all its vertices have even degree, so it can be traced starting from any point.
Student tasks
- Try starting from different vertices and confirm it always works.
Room 4 · The house from the start
The classic house with a roof from the original challenge. Now the student discovers that it does not work from every point: there are special vertices.
Student tasks
- Try tracing from different points.
- Identify from which vertices the trace can be completed.
Room 5 · The house with a door
A house with an extra door line. Only two special points work as starting positions; the student discovers them by experimenting.
Student tasks
- Find the two special points from which the trace can be done.
- Complete the trace starting from one of them.
Room 6 · Vertices and edges
Graph vocabulary is introduced: the points are called vertices and the lines are called edges. The student identifies and counts them on a figure.
Student tasks
- Count how many vertices the figure has.
- Count how many edges it has.
Room 7 · The degree of a vertex
Degree is defined: the number of edges touching a vertex. The student clicks each vertex and sees its degree in real time.
Student tasks
- Click each vertex and read its degree.
- Mentally note which have high degree and which have low degree.
Room 8 · Even and odd
The student classifies the vertices of several figures according to whether their degree is even or odd, consolidating the distinction that will be key to Euler's rule.
Student tasks
- Calculate the degree of each vertex.
- Classify each one as even or odd.
Room 9 · Traverse without repeating an edge
Three figures to trace. One of them has a trick: it looks possible but is not. The student practises and begins to sense the rule.
Student tasks
- Attempt to trace all three figures.
- Discover which one has a trick and why it cannot be done.
Room 10 · More practice
New figures to consolidate the tracing mechanic without repeating edges, with a wider variety of shapes.
Student tasks
- Trace each figure without lifting the pencil or repeating edges.
Room 11 · Can it be done or not?
Prediction quiz: the student marks YES or NO for each figure without tracing it. Sets the stage for discovering the rule that explains everything.
Student tasks
- Look at each figure and decide whether it is traceable.
- Mark YES or NO and check the result.
Room 12 · The magic rule!
Reveal scene: Leonhard Euler discovered nearly 300 years ago that the number of odd-degree vertices determines whether a figure is traceable and from where. The central moment of the lab.
Student tasks
- Watch the animation that reveals Euler's rule.
- Remember the rule: 0 odd → from any point; 2 odd → from one of them; more than 2 → impossible.
Room 13 · Odd-vertex hunters I
Guided practice: the student clicks the odd-degree vertices of each figure to identify them precisely and apply the rule.
Student tasks
- Click all the odd-degree vertices in each figure.
Room 14 · Odd-vertex hunters II
Second round of figures for hunting odd-degree vertices, with more complex graphs that require careful counting.
Student tasks
- Identify and click the odd-degree vertices in each figure.
Room 15 · The open envelope
An open-envelope-shaped figure (traceable with 2 odd vertices). The student must first find the odd-degree vertices and start from one of them.
Student tasks
- Locate the odd-degree vertices before tracing.
- Start from one of them and complete the trace.
Room 16 · The closed envelope
A closed-envelope figure (no odd-degree vertices: traceable from any point and returning to the start). The student applies the rule before attempting it.
Student tasks
- Count the odd-degree vertices and predict whether it is possible.
- Trace the closed envelope returning to the starting point.
Room 17 · House with a rug
A house with a rug line at the base, which adds complexity. The student identifies the odd-degree vertices and starts from one of them.
Student tasks
- Find the odd-degree vertices in the more complete figure.
- Trace starting from an odd-degree vertex.
Room 18 · The bridges of Königsberg
The historical problem: the city of Königsberg has 4 land areas and 7 bridges. Can all of them be crossed exactly once? The student discovers it is impossible.
Student tasks
- Try to cross all 7 bridges without repeating any.
- Confirm it is impossible because there are more than 2 odd-degree vertices.
Room 19 · Move a bridge
Interactive variant: the student moves one of the 7 bridges to see whether that change makes the route possible. New drag-bridge mechanic.
Student tasks
- Mark which bridge to move and choose where to place it.
- Check whether the new configuration satisfies Euler's rule.
Room 20 · A circular route
The student moves as many bridges as needed until all vertices have even degree, achieving a circular route (one that starts and ends at the same point).
Student tasks
- Move bridges until no odd-degree vertex remains.
- Trace the circular route that returns to the starting point.
Room 21 · Another city
A different city with two islands and their bridges. The student applies the rule independently: finds the odd-degree vertices and starts there.
Student tasks
- Identify the odd-degree vertices.
- Trace the route starting from one of them.
Room 22 · Three islands
A city with three islands where four locations (North, South, A and C) have odd degree. The student must move just ONE bridge so that all degrees become even and the walk is circular.
Student tasks
- Find the single bridge move that makes all degrees even.
- Trace the resulting circular route.
Room 23 · Diploma
Closing and congratulations screen. The student receives their diploma as an explorer of Eulerian graphs.
Student tasks
- View the diploma and celebrate completing the walk.
Rooms to project
The most striking ones to show and discuss in class.