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Real cubes


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Question

In the illustration four flat cube-like shapes are shown. Their patterns are drawn with bold black lines. Which of them can you draw without taking your pencil off the paper or going along the same line twice? Which of them can't be drawn in this way?

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Both examples B and C have more than two vertices with odd degree (odd number of lines intersecting). Therefore, these cannot be drawn without retracing a line or lifting one's pencil. However, examples A and D have only vertices of even degree (4, to be exact), and can therefore be drawn without any retracing.

Or rather: A and D have only two vertices with odd degree. So both can be drawn without lifting the pencil or retracing a line (in both cases, you must start at one of the odd vertices and end at the other).

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Both examples B and C have more than two vertices with odd degree (odd number of lines intersecting). Therefore, these cannot be drawn without retracing a line or lifting one's pencil. However, examples A and D have only vertices of even degree (4, to be exact), and can therefore be drawn without any retracing.

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