12. (10 points) The numbers of 100 athletes range from 1 to 100. If each athlete writes down the largest odd factor of their number on the blackboard, what is the sum of all the numbers written on the blackboard?
【Analysis】Since from $1 \sim 100$, the odd multiples of 64 are 1, with the largest odd factor being 1. The odd multiples of 32 are 2 (32, 96), with their largest odd factors being $1, 3$. Similarly, by finding the number of the largest odd multiples of $16, 4, 2, 1$ and their largest odd factors, we can determine the total sum of the numbers.

【Solution】Since from $1 \sim 100$, the odd multiples of 64 are 1, with the largest odd factor being 1.
The odd multiples of 32 are 2, with the largest odd factors being $1, 3$.
The odd multiples of 16 are 3, with the largest odd factors being $1, 3, 5$.
The odd multiples of 8 are 6, with the largest odd factors being $1, 3, 5, 7, 9$.
The odd multiples of 4 are 13, with the largest odd factors being $1, 3, 5, 7, 9 \cdots 25$.
The odd multiples of 2 are 25, with the largest odd factors being $1, 3, 5, 7, \cdots 49$.
The odd multiples of 1 are 50, with the largest odd factors being $1, 3, 5, 7, \cdots 99$.
$$
1+(1+3)+(1+3+5)+(1+3+5+\cdots+11)+(1+3+5+\cdots+25)+(1+3+5+\cdots+49)+(1+3+5+\cdots
$$
99)
$$
\begin{array}{l}
=1+4+9+(1+11) \times[(11-1) \div 2+1] \div 2+(1+25) \times[(25-1) \div 2+1] \div 2+(1+49) \\
\times[(49-1) \div 2+1] \div 2+(99+1) \times[(99-1) \div 2+1] \div 2 \\
=14+36+169+625+2500 \\
=3344 .
\end{array}
$$

Thus, their total sum is 3344.