deg · The Bridges of Königsberg
The Bridges of Königsberg
An island sits in the river, and seven bridges connect the landmasses. Can you take a walk that crosses every bridge exactly once?
The 1736 stroll puzzle that made Euler invent graph theory
Think about it first
Make a prediction first. Change one input at a time and compare what happens with what you expected.
First, tap the land where you want to start
Read the solution when you are ready
The bridge problem is not about the map — it is about dots and lines.
Why does this question matter?
The 18th-century Prussian city of Königsberg (today's Kaliningrad) had an island in its river and seven bridges. A puzzle spread among its citizens: is there a walk that crosses every bridge exactly once?
In 1736, the mathematician Leonhard Euler answered without crossing a single bridge. He erased the shapes of the land and the lengths of the bridges, turning land into dots and bridges into lines. It was a compression that kept only the information the problem actually needed.
On that compressed picture the answer became simple: just count the lines at each dot. That single move started graph theory, the root of all the mathematics we now use for networks.
Read the idea from another angle
Euler didn't cross the bridges — he redrew the map.
Instead of walking hundreds of trials, mathematics changes the representation of the problem. Land size and bridge length are irrelevant to the stroll. Keep only dots and lines, and the answer shows itself.
A single odd dot decides the fate of the whole walk.
At a pass-through dot you must leave as often as you arrive, so it needs an even number of lines. Odd dots are allowed in exactly two places: the start and the end. Königsberg had four, so the walk was impossible.
Now read the definitions with their conditions
Writing the 'stuckness' you experienced in the language of math: the number of lines at a dot is called its degree, and the count of odd-degree dots decides whether the walk exists.
Formula anatomy — matched with what you did in the game
⇔
Degree
One number summarizes a dot's connections. Königsberg's four dots had degrees 5, 3, 3, 3.
Euler's condition
The condition for a path that uses every line exactly once. With 2 odd dots, they must be the start and the end.
Handshake lemma
Each line raises the degree of both its ends by one, so degrees always sum to an even number. That's why odd dots only come in pairs.
Make one more distinction
With the right starting point and order, you could eventually succeed.
It is impossible no matter how often you try. All four landmasses touch an odd number of bridges. At an odd dot, arrivals and departures can't pair up — and with four such dots, even using them as start and end isn't enough.
Stroll puzzles like this are pastimes, not mathematics.
Euler opened a whole new mathematics — graph theory — from this puzzle. Subway maps, the internet, delivery routes, and DNA sequencing are all its descendants. The key was the eye that compresses a problem into dots and lines.
Where else can this idea help?
Subway maps
A transit map erases real geography and keeps only stations and connections — exactly the compression Euler performed on Königsberg.
Delivery and snowplow routes
Finding the shortest route that covers every street once — the route inspection problem — starts from Euler's condition. More odd dots mean more repeated streets.
Networks and circuit design
Internet backbones, power grids, and circuit-board wiring tests are designed and diagnosed with the mathematics of dots and lines.
DNA sequencing
Algorithms that stitch short DNA reads into a genome use Euler paths. The 1736 stroll is at work inside life science.
Explain it in your own words
Check the idea with practice problems (4)
Questions completed 0 / 4
1. To draw every line exactly once without lifting your pen (one stroke), what condition is needed?
2. In the 1736 map of Königsberg, how many landmasses had an odd degree?
3. A figure has exactly 2 odd dots. Where must a one-stroke drawing start?
4. If you build one new bridge (line), what happens to the degrees?
What question comes next?
Previous concept
Fermi Estimation
Compressing messy reality down to the pieces that matter — this time, a map into dots and lines.
← Fermi Estimation labLeads to next
Wolf, Goat and Cabbage
You turned a map into dots and lines; next, turn invisible states into dots.
Go to the Wolf, Goat and Cabbage lab →Related lab
Polygons
A feel for shapes made of points and segments — the base of graph-reading eyes.
Go to the Polygons lab →Related lab
Coordinates
The first compression that turns real-world position into mathematical language.
Go to the Coordinates lab →