Question 1 Can every integer be represented as the sum of the cubes of a finite number of different integers?
The detailed solution is given in reference [1]. The key to solving the problem in the solution lies in the following identity:
$$
\begin{array}{l}
(n+7)^{3}-(n+6)^{3}-(n+5)^{3}+(n+4)^{3}- \\
(n+3)^{3}+(n+2)^{3}+(n+1)^{3}-n^{3} \\
\quad=48 .
\end{array}
$$

A natural question: How is this identity derived? We expect a fixed number of consecutive integers' cubes (algebraic) sum to be a constant, and solving for the coefficients seems not difficult. But is there a more natural way to derive equation (1), or even to generate a series of similar identities?

For this purpose, for any function $f(x)$ defined on $\mathbf{R}$, define the first-order difference of $f(x)$ (with step size $a$) as follows:
$$
\Delta_{a} f(x)=f(x+a)-f(x) .
$$

It is easy to prove the following properties:
Property 1 If $f(x)$ and $g(x)$ are functions defined on $\mathbf{R}$, then
(1) For any $k, l \in \mathbf{R}$, we have
$$
\Delta_{a}(k f(x)+l g(x))=k \Delta_{a} f(x)+l \Delta_{a} g(x) \text {; }
$$
(2) For any $a, b \in \mathbf{R}$, we have
$$
\Delta_{a} \Delta_{b} f(x)=\Delta_{b} \Delta_{a} f(x) \text {. }
$$

Property 2 If
$$
f(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0},
$$
where $b_{0}, b_{1}, \cdots, b_{n} \in \mathbf{R}$ and $b_{n} \neq 0$, then
(1) When $k>n$, $\Delta_{a_{k}} \cdots \Delta_{a_{2}} \Delta_{a_{1}} f(x)=0$;
(2) When $k=n$,
$$
\begin{array}{l}
\Delta_{a_{k}} \cdots \Delta_{a_{2}} \Delta_{a_{1}} f(x)=n!a_{1} a_{2} \cdots a_{n} b_{n} . \\
\text { Let } f(x)=x^{3} \text {. }
\end{array}
$$

By Property 2, we know
$$
\Delta_{1} \Delta_{2} \Delta_{4} f(x)=3!\times 1 \times 2 \times 4=48,
$$

where $\Delta_{1} \Delta_{2} \Delta_{4} f(x)=\Delta_{1} \Delta_{2}\left((x+4)^{3}-x^{3}\right)$
$$
\begin{aligned}
= & \Delta_{1}\left((x+6)^{3}-(x+4)^{3}-(x+2)^{3}+x^{3}\right) \\
= & (x+7)^{3}-(x+6)^{3}-(x+5)^{3}+ \\
& (x+4)^{3}-(x+3)^{3}+(x+2)^{3}+ \\
& (x+1)^{3}-x^{3} .
\end{aligned}
$$

Let $x=n$, and we get equation (1).
We can further generalize the problem 1: