"Pre-assembled" unit cube has a base and a top, as well as its sides. The top is rotated by an acute angle $\alpha$ around the line connecting the centers of the two faces, relative to the base. By how much does the top get closer to the base if the side edges remain the same length and the connection between the vertices also remains unchanged?
The height of the cube before unscrewing is $m=1$, and the diagonal length of the square plates is $\sqrt{2}$.

After the unscrewing (only the top plate rotates), the projections of the rotated points $A^{\prime}$ and $D^{\prime}$ onto the base plane are denoted as $B^{\prime}$ and $C^{\prime}$, respectively, and the center of the base square is $O$ (Figure 1).

In the top view of the original and rotated squares (Figure 2), $O B=O B^{\prime}=\frac{\sqrt{2}}{2}, B B^{\prime}=x$. In the isosceles triangle $O B B^{\prime}$, $\angle B O B^{\prime}=\alpha$ and $\sin \frac{\alpha}{2}=\frac{\frac{x}{2}}{\frac{\sqrt{2}}{2}}$, from which $x=\sqrt{2} \sin \frac{\alpha}{2}$.

Since the side edges do not change in length during the unscrewing, according to the notation in Figure 1, $A^{\prime} B=1, A^{\prime} B^{\prime}=m^{\prime}$ is the new height, and from the right triangle $A^{\prime} B B^{\prime}$: $m^{\prime 2}+x^{2}=1$, from which

$$
m^{\prime}=\sqrt{1-2 \sin ^{2} \frac{\alpha}{2}}=\sqrt{1-2 \frac{1-\cos \alpha}{2}}=\sqrt{\cos \alpha}
$$

The difference between the original and the new height gives how much closer the top plate has come to the base plate:

$$
m-m^{\prime}=1-\sqrt{\cos \alpha}
$$

Remark. The result also reveals something else. If $\alpha=0$, the value of $\cos \alpha$ is 1, so if nothing is unscrewed, the distance does not change. The unscrewing has a maximum, and this is $90^{\circ}$. In this case, if we imagine the unscrewing, the previously "vertical" (perpendicular to the base plane) edges will lie in the base plane, and the base plate and the top plate will now lie on top of each other.

Várallyay György (Budapest, Kodály Z. Hungarian Chorus, 12th grade)
![](https://cdn.mathpix.com/cropped/2024_05_02_5d1e04bd74043e79a8efg-1.jpg?height=1008&width=574&top_left_y=1083&top_left_x=754)