Teacher Notes: There is a famous problem in Discrete Mathematics called ‘The Bridges of Konigsberg’ in which it is said to be impossible to cross each bridge in the town of Konigsberg, in Russia, once and only once if you wish to return to your start point (you may start anywhere). This problem is mathematically very similar to the problem of trying to draw a shape without taking your pen of the page and without retracing any line you have already drawn.

Don’t take your pen off the page! Draw the diagram of the envelope below without going over the same line twice and without taking your pen off the paper. To show that you have done it you must show your starting and finishing points clearly and mark out your route. A

Answers: It is possible to do if you start at any corner (vertex) that has an ODD number of lines touching it (i.e. you must start at the bottom right or bottom left hand corner). You will always finish at the other ‘odd vertex’.

Don’t take your pen off the page! Draw the diagram of Pythagoras’ Theorem below without going over the same line twice and without taking your pen off the paper. To show that you have done it you must show your starting and finishing points clearly and mark out your route. B

Answers: The only ‘vertices’ with an odd number of ‘edges’ (lines) coming from them are shown in red here: So, providing you start at one of those two ‘vertices’, you will finish at the other one.

Don’t take your pen off the page! Draw the diagram of the ‘Star of David’ below without going over the same line twice and without taking your pen off the paper. To show that you have done it you must show your starting and finishing points clearly and mark out your route. C

Answers: Because this ‘Star of David’ doesn’t have any ‘odd vertices’ you can start and finish anywhere – it’s easy! (Or easier anyway). Incidentally, if you have any picture with more than two ‘odd vertices’ it is impossible to draw without taking your pen of or retracing your steps (hence walking the bridges of Konigsberg is impossible unless you retrace your steps or teleport from one section to another!)