What is the largest natural number $n$ for which the rectangular parallelepipeds $T_{1}, T_{2}, \ldots, T_{n}$ can be specified such that their faces are parallel to one of the coordinate planes and for which it holds that for any subset $H$ of the set of the first $n$ natural numbers, there exists a point $P$ such that $P$ is an element of $T_{i}$ if and only if $i$ is an element of $H$ $(i=1,2,3, \ldots, n)$.
A rectangular parallelepiped's $(x, y, z)$ points in space can be characterized by the conditions

$$
a \leqq x \leqq b, \quad c \leqq y \leqq d, \quad e \leqq z \leqq f
$$

Accordingly, for each rectangular parallelepiped, there is an interval on each coordinate axis. If the system $T_{1}, T_{2}, \ldots, T_{n}$ is appropriate, then there is a point $(x, y, z)$ that lies within each $T_{i}$ rectangular parallelepiped. Let $A$ be the largest of the lower bounds of the intervals corresponding to the first coordinates (or one of them), and let $T_{A}$ be the corresponding parallelepiped. Similarly, we can define the numbers $B, C, D, E, F$ and the parallelepipeds $T_{B}, \ldots, T_{F}$ (the latter may not necessarily be distinct). It can be seen that $n \leqq 6$, since, by the definition of the quantities $A, B, C, D, E, F$, the intersection of the parallelepipeds $T_{A}, T_{B}, T_{C}, T_{D}, T_{E}, T_{F}$ would be a subset of any possible additional parallelepiped. (Here, $B$ is the smallest of the upper bounds of the intervals corresponding to the first coordinates.)

|  | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $a$ | -1 | -2 | -3 | -3 | -3 | -3 |
| $b$ | +2 | +1 | +3 | +3 | +3 | +3 |
| $c$ | -3 | -3 | -1 | -2 | -3 | -3 |
| $d$ | 3 | 3 | 2 | 1 | 3 | 3 |
| $e$ | -3 | -3 | -3 | -3 | -1 | -2 |
| $f$ | 3 | 3 | 3 | 3 | 2 | 1 |

We demonstrate with an example that $n=6$ is achievable. The data for the parallelepipeds are given in the accompanying table. As can be seen, $A=C=E=-1, B=D=F=1, T_{A}=T_{1}, T_{B}=T_{2}$, etc. The accompanying diagram shows the regions corresponding to the different subsets in the layer $2<z<3$ of the space. Below this, in the layer $1<z<2$, are the regions where the number 5 is added to the sets, and below that, in the layer $-1<z<1$, the number 6 is also added to the sets, except for the number 5. Finally, in the layer $-2<z<-1$, the number 5 exits the sets. Ultimately, we found a region in each of the 64 subsets, so the construction is appropriate.

![](https://cdn.mathpix.com/cropped/2024_05_02_7b75e8e8f3bf16d819c3g-1.jpg?height=884&width=876&top_left_y=1190&top_left_x=619)