1. Peter wrote several 1s on a piece of paper, inserted the symbol “+” or “$\times$” between each pair of 1s, and added some parentheses to make the result 2014. Basil changed all the “$\times$” to “+” and all the “+” to “$\times$”, and also got 2014. Is this possible?
1. This is indeed possible.

For example, Peter writes
$$
1+\underbrace{1 \times 1+\cdots+1 \times 1}_{2013 \uparrow}=2014 \text {. }
$$

Balthazar changes what Peter wrote to
$$
\underbrace{1 \times 1+1 \times 1+\cdots+1 \times 1}_{2013 \uparrow}+1=2014 .
$$