The faces of a cube are each painted either red or blue.
An ant is positioned on each of the eight corners of the cube, and each ant can only see the three faces that meet at that corner of the cube.
In this question, the ants will be asked about the faces that they can see, and the ants will always answer truthfully.
(i) The ants are each asked "Can you see an even number of red faces?". Each ant answers either yes or no. Explain why the number of ants that say yes is even.
(ii) Is it possible that all eight of the ants can each see exactly two red faces? Justify your answer.
(iii) The ants are each asked "Can you see at least one red face?". Explain why it is impossible for exactly five of the ants to say yes and exactly three to say no.
(iv) Suppose that the four ants on the corners of the top face of the cube can see exactly $0,1,1$, and 2 red faces each, in some order. How many blue faces might there be in total? Find all possibilities, and explain your answer.
(v) A three-dimensional shape is constructed such that each face is either a square or a hexagon, with two faces meeting at each edge and three faces meeting at each corner. Each face of the shape is painted either red or blue. Consider the edges where a red face meets a blue face. Explain why the number of such edges is even.