Five. (25 points) In the expression “ $\square 1 \square 2 \square 3 \square 4 \square 5 \square 6 \square 7$ $\square 8 \square 9$ ”, fill in the small squares with “+” or “-” signs. If the algebraic sum can be $n$, then the number $n$ is called a "representable number"; otherwise, it is called an "unrepresentable number" (for example, 1 is a representable number, because
$$
+1+2-3-4+5+6-7-8+9
$$

is one way to represent 1).
(1) Prove that 7 is a representable number, and 8 is an unrepresentable number;
(2) Find the number of different ways 25 can be represented.

保留源文本的换行和格式，直接输出翻译结果。
(1) Since $+1-2-3+4+5-6+7-8$ $+9=7$, therefore, 7 is a number that can be represented.
$$
\text { Also, }+1+2+3+4+5+6+7+8+9=45
$$
is an odd number, and for any two integers $a$ and $b$, $a+b$ and $a-b$ have the same parity. Therefore, no matter how the “ $+\cdots \times$ - ” signs are filled, the algebraic sum must be an odd number, and it cannot be 8. Therefore, 8 is a number that cannot be represented.
(2) Let the sum of the numbers with “+” signs be $x$, and the sum of the numbers with “-” signs be $y$. Then $x-y=25$.
$$
\begin{array}{l}
\text { Also, } x+y=1+2+\cdots+9=45 \text {, then } \\
x=35, y=10 .
\end{array}
$$

Since $9<10,1+2+3+4=10$, therefore, the numbers with “-” signs must be at least 2 and at most 4.
Thus, the sum of the numbers with “-” signs is 10.
Next, we just need to place a “-” sign in front of the numbers whose sum is 10, and a “+” sign in front of the other numbers to get one way to represent 25. Therefore, we only need to calculate the number of different ways to select several numbers from 1 to 9 whose sum is 10, to get the number of different ways to represent 25.
(i) 10 equals the sum of two numbers: $10=1+9=2+8=$ $3+7=4+6$, a total of 4 methods;
(ii) 10 equals the sum of three numbers: $10=1+2+7=1+$ $3+6=1+4+5=2+3+5$, a total of 4 methods;
(iii) 10 equals the sum of four numbers: $10=1+2+3+4$, only 1 method.

In summary, there are $4+4+1=9$ different ways to represent 25.
(Provided by the Sichuan Mathematical Competition Committee)