- Build an object as follows. Choose the first piece (the it piece). You can only make connections to the it piece. After each connection, the piece that is connected to itbecomes the new it (this can be a new piece or one that is already part of the object). Continue until you cannot make any new connections ("it" is already connected on all sides). Can you build the entire object (make all connections) in this manner?
For example, in the picture at right, assume Red is it. After connecting it to Green,
Green becomes it. After the connection Green-Yellow, Yellow becomes it, and then Blue.
If we connect Blue to Green, Green is it again and we need to add a new piece to Green, etc.
Hint: always keep it in your hand and make the connections with your other hand.
Tetrahedron: no, cube: yes, octahedron: no, triangular prism: yes,
dodecahedron: no, truncated octahedron: yes, etc.
Does it matter:
- which piece you choose as your starting it, and
- in which order you make the connections?
Answer: (1) cube: no, triangular prism: yes, truncated octahedron: no, etc. (2) yes.
Hint: this is the famous
"Seven Bridges of Königsberg"
problem in graph theory; an object can be built in this way if the graph that has as nodes the object faces has an
- Can you predict beforehand which objects can be completed this way?
Hint: you must now find a
, a much harder problem.
- Change the game so that a new it can only be a new piece that you're adding (make all connections to the old it before moving to the new one). Which objects can you build now?
Platonic solids - yes, cuboctahedron, icosidodecahedron - no (prove it!), etc.