Question 6
Consider all tickets bearing nine-digit numbers formed from the digits 1, 2, 3. Two tickets are called “separated” if their numbers $\overline{a_1a_2\cdots a_9}$ and $\overline{b_1b_2\cdots b_9}$ satisfy $a_i\ne b_i$ for every $i=1,2,\ldots,9$.
(a) Prove that all tickets can be coloured blue, red or yellow, one colour per ticket, so that separated tickets always have different colours.
(b) Consider any colouring satisfying (a). Suppose ticket 122222222 is red and ticket 222222222 is yellow. What colour is ticket 123123123?
(a) Prove that all tickets can be coloured blue, red or yellow, one colour per ticket, so that separated tickets always have different colours.
(b) Consider any colouring satisfying (a). Suppose ticket 122222222 is red and ticket 222222222 is yellow. What colour is ticket 123123123?