Question 69
Integers between $1$ and $7$ are written on a blackboard. Not all numbers from $1$ to $7$ need be present, and one, several or all numbers may be repeated one or more times.
A move consists of choosing one or more numbers on the board, all distinct, erasing them, and replacing them with the numbers which together with those erased form the entire set $\{1,2,3,4,5,6,7\}$.
For example, allowed moves are:
- Erase one $4$ and one $5$, and replace them with $1,2,3,6,7$;
- Erase one each of $1,2,3,4,5,6,7$ and write nothing.
Prove that if a sequence of moves from the initial position can leave a single number on the board (written just once), then that number does not depend on the sequence of moves used.
A move consists of choosing one or more numbers on the board, all distinct, erasing them, and replacing them with the numbers which together with those erased form the entire set $\{1,2,3,4,5,6,7\}$.
For example, allowed moves are:
- Erase one $4$ and one $5$, and replace them with $1,2,3,6,7$;
- Erase one each of $1,2,3,4,5,6,7$ and write nothing.
Prove that if a sequence of moves from the initial position can leave a single number on the board (written just once), then that number does not depend on the sequence of moves used.