14. 2011 cards are arranged in a row on a table. One of the numbers ' 1 ', ' 2 ' and ' 3 ' is printed on each card. It is found that there is at least one card between any two cards labelled ' 1 ', at least two cards between any two cards labelled ' 2 ', and at least three cards between any two cards labelled ' 3 '. If the smallest and greatest possible numbers of cards labelled ' 3 ' are $m$ and $M$ respectively, find the value of $m+M$.
2011 cards are placed in a row on a table, each card has one of the numbers '1', '2', or '3' written on it. It is known that there is at least one card between any two cards labelled '1', at least two cards between any two cards labelled '2', and at least three cards between any two cards labelled '3'. If the smallest and greatest possible numbers of cards labelled '3' are $m$ and $M$ respectively, find the value of $m+M$.
14. 1005
14. Consider any 4 consecutive cards. There can be at most one '3' since there are at least three cards between any two '3's. On the other hand, there is at least one '3', for otherwise they have to be '1's and '2's subject to the required conditions, which can easily be seen to be impossible. In other words, there is exactly one '3' among any 4 consecutive cards.
Since $2011 \div 4=502.75$, the number of '3's must either be 502 or 503. Both bounds can be achieved (for the former case, consider 12131213...1213121; for the latter case, consider $31213121 \ldots 3121312$). Hence $m+M=502+503=1005$.