Can you draw this shape in one stroke?
Can you draw it in one stroke?
Use every line exactly once without lifting your pencil.
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Click a vertex to start, then click a neighbouring vertex to move.
One-stroke drawings are Euler trails
When can a graph be drawn in one stroke?
odd vertices
✓ One stroke
Start and finish at the same vertex.
odd vertices
✓ One stroke
Start at one odd vertex and finish at the other.
odd vertices
✗ Impossible to draw every edge exactly once in one stroke.
The house
Look at the graph before trying to draw it.
Can it be drawn in one stroke? If so, where must you start and finish?
Count the degree of each vertex before you reveal the answer.
Does a crossing count?
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Learning tip
Before counting degrees, decide exactly where the vertices are.
Two edges crossing on the page does not necessarily mean there is a
vertex there. Mark the vertices first, then count the edges meeting them.
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This square has both diagonals drawn across it.
Can you draw the whole graph in one stroke?
Count the degree of each vertex carefully. What about the place where the two diagonal lines cross?
Hint: vertices are marked by dots.
Challenge — Fix the drawing
The crossed square has four odd-degree vertices, so it cannot be drawn in one stroke.
You may add one extra edge between two corners, even if those corners are already joined. Draw the new edge as a separate curved line.
Can you make the new drawing possible to complete in one stroke?
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Reveal answer
Yes.
Join any two of the four odd-degree corners with one extra edge.
Adding an edge changes the degree of both of its endpoints by 1. Those two odd degrees therefore become even.
The graph is left with exactly two odd-degree vertices, so it now has an Euler trail.
The one-stroke drawing must start at one of the two remaining odd vertices and finish at the other.
One possible extra edge is shown in purple.
What have we learned?
One-stroke drawing puzzles are really Euler-trail problems.
Instead of trying route after route, count the odd-degree vertices first.
Zero odd vertices: possible, finishing where you started
Two odd vertices: possible, starting and finishing at the odd vertices
More than two odd vertices: impossible
Come back to it later
When you learn something new, it is important to come back to it later. Revisiting an idea after some time has passed helps you discover what you remember, what needs another look, and often makes the idea clearer the second time around. Our handouts are a perfect excuse to do exactly that.
Printable Graphs course
Prefer paper, a tablet, or studying offline? Open the printable version of the whole Graphs course, including this lesson.
One-stroke drawings handouts
Get the printable Lesson 4 handout and the accompanying set of supplementary examples.
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