
6. In a lottery, tickets are given nine-digit numbers using only the digits $1,2,3$. They are also coloured red, blue or green in such a way that two tickets whose numbers differ in all the nine places get different colours. Suppose
the ticket bearing the number 122222222 is red and that bearing the number 222222222 is green. Determine, with proof, the colour of the ticket bearing the number 123123123 .


Solution: The following sequence of moves lead to the colour of the ticket bearing the number 123123123:

| Line Number | Ticket Number | Colour | Reason |
| :---: | :--- | :--- | :--- |
| 1 | 122222222 | red | Given |
| 2 | 222222222 | green | Given |
| 3 | 313113113 | blue | Lines 1 \& 2 |
| 4 | 231331331 | green | Lines 1 \& 3 |
| 5 | 331331331 | blue | Lines 1 \& 2 |
| 6 | 123123123 | red | Lines 4 \& 5 |

If 123123123 is reached by some other root, red colour must be obtained even along that root. For if for example 123123123 gets blue from some other root, then the following sequence leads to a contradiction:

| Line Number | Ticket Number | Colour | Reason |
| :---: | :--- | :--- | :--- |
| 1 | 122222222 | red | Given |
| 2 | 123123123 | blue | Given |
| 3 | 231311311 | green | Lines 1 \& 2 |
| 4 | 211331311 | green | Lines 1 \& 2 |
| 5 | 332212212 | red | Lines 4 \& 2 |
| 6 | 113133133 | blue | Lines 3 \& 5 |
| 7 | 331331331 | green | Lines 1 \& 2 |
| 8 | 222222222 | red | Line 6 \& 7 |

Thus the colour of 22222222 is red contradicting that it is grren.

