[Example 4.5.1] Nine bags contain $9,12,14,16,18,21,24,25,28$ balls respectively. A takes several bags, B also takes several bags, and in the end, only one bag is left. It is known that the total number of balls A takes is twice the total number of balls B takes. How many balls are in the remaining bag?
The known conditions of this problem are three: The first condition is the number of balls in 9 bags; the second condition is that Jia and Yi together took the balls from 8 bags; the third condition is that the number of balls Jia took is twice the number of balls Yi took.

Considering these known conditions one by one according to the requirements of the problem, there are many situations that simultaneously meet the three conditions: There are $C_{9}^{8}=9$ ways to choose 8 bags out of 9; and for each selection of 8 bags, dividing them into two groups, one for Jia and one for Yi, there are $C_{8}^{1}+C_{8}^{2}+\cdots+C_{8}^{7}=2^{8}-2=254$ ways to divide, so the total number of ways to meet the second condition is $9 \times 254=2286$. Verifying whether each of these 2286 cases meets the third condition by enumeration is obviously impractical.
However, if we think about the known conditions as a whole, the situation is different.
If we let Jia take a total of $x$ balls and Yi take a total of $y$ balls, then by the third condition, $x=2y$, so $x+y=3y$.

From the overall perspective of the total number of balls taken by Jia and Yi, the sum of the balls they took is a multiple of 3.
Thus, we can think about the total sum of the 9 bags of balls as a whole.
The total sum of these 9 bags of balls is
$$
\begin{aligned}
& 9+12+14+16+18+21+24+25+28 \\
\equiv & 0+0+2+1+0+0+0+1+1 \\
\equiv & 2(\bmod 3) .
\end{aligned}
$$

This indicates that the sum of the 9 bags of balls is congruent to 2 modulo 3, while the sum of the 8 bags of balls is a multiple of 3, so the number of balls in the remaining bag must be congruent to 2 modulo 3. By examining the remainder of each of the 9 bags of balls when divided by 3, only 14 leaves a remainder of 2 when divided by 3.
The answer to the problem is thus that the number of balls in the remaining bag is 14.