954. Solve the inequality in natural numbers $x$ and $y$

$$
x^{2}+1<y^{2}<x^{2}+x+1
$$
$\triangle$ The inequality holds

$$
x^{2}<x^{2}+1<y^{2}<x^{2}+x+1<x^{2}+2 x+1=(x+1)^{2}
$$

From this,

$$
x^{2}<y^{2}<(x+1)^{2}, \quad x<y<x+1
$$

But the last inequality has no solutions in natural numbers $x$ and $y$. Answer: no solutions.

Here we applied the "pinching" method, familiar to us from § 20.