# Why are these shapes impossible to draw with the given rules?

The image above shows a group of shapes, all with lines leading to their center point. These shapes cannot be drawn with the following rules:

1. You may not trace over over an already existing line.
2. You may only draw lines that are part of the shape.
3. You may not move your writing utensil while it is up and you are drawing this shape.

This can be done within shape, not just the above. Why is this?

• Graph theory 101. – Ian MacDonald Jul 13 '16 at 19:25
• Not a hard/good puzzle, but a very valid question for somebody honestly seeking reasons and explanations. I don t think it deserves DV just because it is easy to answer, as this is a WHY question, not a 'Solve my challenge' puzzle. – BmyGuest Jul 13 '16 at 19:56
• @BoxTechy When you think a posting answerd your questions, it would be courtesy to 'accept' the according answer with the little checkbox item under the voting buttons. – BmyGuest Jul 13 '16 at 21:32
• Essentially a duplicate of Drawing something using one pen stroke. – Peregrine Rook Jul 28 '16 at 10:52

This is a variation of the Seven Bridges of Königsberg puzzle.

The answer is you can't trace out the pattern if more than two nodes are odd. We call a node "odd" if it has an odd number of lines to it, and "even" if an even number. (If a node is not an endpoint of the drawn line, then the drawn line must enter that node the same number of times as it leaves it, so the node must be even. So the only nodes that can be odd are the endpoints of the drawn line. Because a single line has two endpoints, a beginning and an end, more than 2 odd nodes would indicate more than 2 endpoints.)

• You have a typo -- "if more than two nodes ..." should be "if no more than two nodes". – Gareth McCaughan Jul 13 '16 at 18:52
• @GarethMcCaughan fixed. – Jiminion Jul 13 '16 at 20:03