Student investigation booklet
Mathematical
Mysteries
Join Magda to simplify maps, trace magic paths, defend networks and justify why your solution must work.
Magda’s Network Words
A network keeps the connections and removes detail that does not matter.
Sketch one symbol for each word as it appears in the online story.
A place or point
One vertex; two or more vertices.
A connection
A road, bridge or link between two vertices.
Edges meeting here
The degree of a vertex is the number of edges that meet it.
A journey through edges
A one-stroke trail uses every edge exactly once.
Every place can be reached
A network is connected when you can travel along edges from any vertex to every other vertex.
1Simplify an everyday network
Choose a school, park, transport or family network.
Vertices could represent
Edges could represent
Places Become Points
Connection matters more than the position of a dot.
1Copy the connections
Use the digital Network Builder. Draw your network here with a different layout, but keep every connection the same.
2Record the degrees
Label the vertices A, B, C and D.
| Vertex | Degree | Odd or even? |
|---|---|---|
| A | ||
| B | ||
| C | ||
| D |
Total number of edges: ______
3Same network or different network?
Look at these diagrams. Decide whether the connections are the same.
Network X
Network Y
I know because
Predict → Test → Notice
Trace every edge exactly once. Do not lift your pencil or retrace an edge.
First mark a predicted start. Then trace. If you get trapped, keep the attempt visible and record where it happened—failed routes contain useful evidence.
Predicted start: ______ Finish: ______
Possible? □ Yes □ No
Predicted start: ______ Finish: ______
Possible? □ Yes □ No
Where will I start?
Where did I travel?
Where could I get trapped?
What will I change?
1Count before you trace again
| Network | Vertices with even degree | Vertices with odd degree | Successful start(s) |
|---|---|---|---|
| Mystery A | |||
| Mystery B |
I think the start matters when
Can Every Network Work?
Use degree evidence to predict before tracing.
Odd vertices: __________________________
Prediction: □ possible □ impossible
Odd vertices: __________________________
Prediction: □ possible □ impossible
1Test only after predicting
Trace both. If a network is impossible, write why. If it is possible, record the starts that can work.
My rule idea: A connected network can have a one-stroke trail when the number of odd-degree vertices is ____________________. If there are two odd vertices, the route must start and finish ________________________________.
2Use the degree clue
Mystery C cannot work because it has odd vertices.
Mystery D must start at or because these are its odd vertices.
The Seven Bridges Mystery
Keep the connections; remove the distracting map detail.
Can one route cross all seven bridges exactly once? Turn each land region into a vertex and keep each bridge as an edge.
| Land-region vertex | N | S | C | D |
|---|---|---|---|---|
| Degree | ||||
| Odd or even? |
1Make the decision before tracing
My degree evidence is
2What stayed the same?
The pictures look different, but every land region and bridge still has a matching point or edge.
The route problem stays the same because
Odd, Even and Why
Organise the evidence from several networks before stating a rule.
| Network | Number of odd-degree vertices | One-stroke trail? | Where can it start and finish? |
|---|---|---|---|
| Mystery A | |||
| Mystery B | |||
| Mystery C | |||
| Mystery D | |||
| Seven Bridges |
Why pairs matter
At a vertex in the middle of a trail, one edge brings you in and another edge takes you out. Draw pairs around this degree-4 vertex.
Why odd vertices are special
At an odd vertex, one edge is left unpaired. That unpaired edge can belong at a route’s start or finish. Draw pairs around this degree-3 vertex.
Complete the rule. In a connected network, a one-stroke trail is possible when there are _____ or _____ odd-degree vertices. With no odd vertices, a trail can finish where it starts. With two odd vertices, it must start at one odd vertex and finish at the other.
Why does the rule work?
The pairing idea works because
Protect the Network
Predict what becomes unreachable before removing a connection.
An important edge is a connection whose removal splits the network. An important vertex is a place whose removal splits the remaining network.
1Edge attack
Circle one edge whose removal would split the town into two groups.
I chose edge ______ because:
2Vertex attack
Cross out one vertex whose removal would isolate at least two other vertices from each other.
I chose vertex ______ because:
3Make the smallest repair
Add exactly one new edge to the town network so that removing edge B–D no longer splits it. Label your edge, then explain the alternate route.
| Before my new edge | After my new edge |
|---|---|
| Places reachable from A if B–D fails: __________________ | Places reachable from A if B–D fails: __________________ |
Colour the Map
Regions that share a boundary must differ. Touching at one point does not count.
Use no more than four colours. If you prefer patterns, use blank, dots, stripes and crosses. Write a colour or pattern key before you begin.
1Check
Tick each shared boundary after the regions on its two sides have different colours or patterns.
I found ______ shared boundaries.
2Find the smallest number
Colour ring A–E first. The smallest number that works for the ring is ______.
F touches all five. Can F reuse a ring colour? □ Yes □ No
Smallest number for the whole map: ______
How Many Do We Really Need?
Using three colours does not yet show that two colours are impossible.
Each line means the two regions share a boundary.
1Show that three are needed
A touches B. B touches C. C also touches A.
Could all three use one colour? □ Yes □ No
Could all three use only two colours? □ Yes □ No
Smallest number that works: ______
Explain your answer
A, B and C need three colours because every region
2Point or boundary?
For each pair, decide whether the regions are neighbours under the map-colouring rule.
P and Q: □ neighbours □ not neighbours
R and S: □ neighbours □ not neighbours
Accuracy check: Today you validate examples and justify why a particular group needs a certain number of colours. You are not proving the full Four-Colour Theorem.
Create a Mathematical Mystery
Design it, solve it yourself, then invite someone else to test it.
1Write precise rules
The solver must
2Show why your answer works
My answer works because
Creator check
- I solved my own puzzle.
- The rules allow one clear decision.
- My explanation uses a property.
Tester note
What was clear? ________________________
What needs revision? ___________________
Tester initials: ______
Think Like a Mathematician
Use the evidence in your booklet to make your final claims.
1A choice that changed the whole network
Describe one start, edge, vertex or colour choice and its later consequence.
2Maths of connections
Where might a person use network mathematics outside this workshop?
3Final justification
I claim
My evidence is
Therefore
One question I still have: ______________________________________________________________________________________