IT-CESENATICO 2011 Problem 3

Integers between $1$ and $7$ are written on a blackboard. It is possible that not all the numbers from 1 to 7 are present, and it is also possible that one, some or all of the numbers are repeated, one or more times. A move consists of choosing one or more numbers on the blackboard, where all distinct, delete them and write different numbers in their place, such that the written numbers together with those deleted form the whole set $\{1, 2, 3, 4, 5, 6 , 7\}$ For example, moves allowed are: • delete a $4$ and a $5$ , and write in their place the numbers $1, 2, 3, 6$ and $7$ ; • deleting a $1$ , a $2$ , a $3$ , a $4$ , a $5$ , a $6$ and a $7$ and write nothing in their place.

Prove that, if it is possible to find a sequence of moves, starting from the initial situation, leading to have on board a single number (written once), then this number does not depend on the sequence of moves used.