Where's the maths? Strip away the houses, the river and the distances and only the
connections remain: which pieces of land are joined, and by how many bridges. A picture
of dots and lines like that is a graph, and the study of graphs is graph theory
— the mathematics of networks, from bus routes and molecules to the internet. Euler's
insight was that a simple counting fact (how many dots have an odd number of lines)
answers the walking question completely, with no need to try every route. That is a
proof of impossibility, something no amount of experimenting could ever give you.
📘 On the Sec 1–4 syllabus
- uses Sec 1 · N1 Numbers and their operations — odd and even numbers
Beyond the syllabus: graph theory is enrichment. It appears in no O-Level topic, which is part of why it is fun.
In 1736 the people of Königsberg asked a holiday question: could you walk through
the city and cross every one of its seven bridges exactly once? Leonhard Euler proved
it was impossible — and invented graph theory while doing it.
Throw the map away. The only thing that matters is which land masses are joined:
4 dots (vertices) and 7 lines (edges). The degree of a vertex is the
number of bridge-ends touching it.
Every time your walk enters a land mass it must also leave it, so bridges
get used up in pairs. A land mass in the middle of the walk therefore needs an even
degree. Only the start and the finish are allowed to be odd.
A connected graph has an Euler path only if it has 0 or 2 odd vertices, and an Euler
circuit (finish where you started) only if it has 0.
Königsberg's degrees are 3, 3, 3 and 5 — four odd vertices, so no route exists.
No amount of cleverness helps; it is the parity that forbids it.
Handshake lemma: every bridge has two ends, so the degrees always add up to twice the
number of bridges: 3+3+3+5 = 14 = 2 × 7. That is why the number of odd vertices is always
even. It is the same idea behind "draw this figure without lifting your pen".
Try this: switch to Edit bridges and delete one of the two
bridges on the island's west side. Watch the degrees change to 2, 3, 4, 3 — two odd
vertices. Now press Show a route. Which land mass does the walk have to start on,
and why can't it start anywhere else?