What it shows
Four land areas joined by seven bridges cannot be walked in one trip crossing every bridge once, because every land area has an odd number of bridges. Lifting any one bridge off the board leaves only two odd areas and the walk becomes possible, starting at one odd area and ending at the other.
The science
Shrink each land area to a dot and each bridge to a line between dots: this is a graph. Walking a route that uses every line once enters and leaves each dot in pairs, so every dot except the start and end needs an even number of lines. In Konigsberg the four dots have 3, 3, 3 and 5 lines, four odd dots, so no such walk exists. Euler proved in 1736 that a walk exists exactly when there are 0 or 2 odd dots, and a closed loop exists when there are none. Removing a bridge changes two dots' counts by one, leaving two odd dots.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A board painted as a river with two banks and two islands
- Seven removable card bridges with velcro or pegs
- A toy figure to walk the route
- A dry-erase sheet to draw the graph
- Labels A to D for the land areas
Safety
This project uses safe, everyday materials. Still, read the rules below with a parent before you start.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Paint the board: a river running left to right, a north bank, a south bank, a large island in the middle and a smaller island to the right.
Fix the seven bridges: two from the north bank to the big island, two from the south bank to the big island, one from each bank to the small island, and one between the two islands.
Invite visitors to walk the toy across every bridge exactly once and keep a tally of attempts; none will succeed.
Draw the graph on the dry-erase sheet with four dots and seven lines, and write the number of lines at each dot: 3, 3, 3, 5.
Explain the rule: a walk crossing every bridge once needs at most two dots with an odd count.
Remove one bridge between the north bank and the big island so the counts become 2, 3, 3, 5, wait, check again: removing it lowers both ends by one, giving 2, 3, 2, 5 for north bank, south bank, small island and big island... correct the sheet to the actual new counts and show two odd dots remain.
Walk the toy along a route starting at one odd dot and ending at the other to prove it can now be done.
Add an eighth bridge instead and show that with 2, 4, 4, 4 or similar even counts a closed loop also works.
What you should see
No visitor completes the seven-bridge walk, the graph shows four odd dots, and with one bridge removed the toy completes a walk starting and ending at the two odd dots.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why must the middle dots of a walk have an even count?
Each visit uses one line to enter and one to leave, so lines come in pairs.
2How many odd dots does the Konigsberg graph have?
Four, with counts 3, 3, 3 and 5.
3What did Euler show?
That such a walk exists if and only if there are zero or two odd dots, which started the subject of graph theory.
4Where is this used today?
In planning routes for postal delivery, garbage collection and snow ploughing, where every street must be covered once.
Take it further
Draw the map of your school corridors as a graph and find a route that walks every corridor exactly once, or prove there is none.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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