How's the IUPAC name of basketane determined?
Validation
Let us build this alien step by step:

We will build bicyclo[$4$.$4$.$0$]decane.
We will build tricyclo[$4$.$4$.$0$.$\mathrm{0^{2,5}}$]decane.
We will build tetracyclo[$4$.$4$.$0$.$\mathrm{0^{2,5}}$.$\mathrm{0^{3,8}}$]decane.
We will build pentacyclo[$4$.$4$.$0$.$\mathrm{0^{2,5}}$.$\mathrm{0^{3,8}}$.$\mathrm{0^{4,7}}$]decane.


Bicyclo[$4$.$4$.$0$]decane looks like this:

The part &quot;decane&quot; means that there are $10$ carbons and the molecule is saturated, just as indicated.
The part &quot;bicyclo&quot; means that there are two rings. One ring is $1$-$2$-$3$-$4$-$5$-$6$ and the other is $1$-$6$-$7$-$8$-$9$-$10$.
The part [$4$.$4$.$0$] is based on $\ce{C}1$ and $\ce{C}6$:

The first $4$ refers to the first link between $\ce{C}1$ and $\ce{C}6$, namely $2$-$3$-$4$-$5$ which contains $4$ carbons.
The second $4$ refers to the second link between $\ce{C}1$ and $\ce{C}6$, namely $7$-$8$-$9$-$10$ which contains $4$ carbons
The $0$ refers to the third link between $\ce{C}1$ and $\ce{C}6$ which is just a line with $0$ carbons between.


Tricyclo[$4$.$4$.$0$.$\mathrm{0^{2,5}}$]decane looks like this:

It is now &quot;tricyclo&quot; because there are three rings: $1$-$2$-$5$-$6$, $2$-$3$-$4$-$5$, $1$-$6$-$7$-$8$-$9$-$10$.
The extra part $0^{2,5}$ means that $\ce{C}2$ and $\ce{C}5$ are linked by $0$ extra carbons.

Tetracyclo[$4$.$4$.$0$.$\mathrm{0^{2,5}}$.$\mathrm{0^{3,8}}$]decane looks like this:

(from now on, a strong intuition on topology is required)
The new part $\mathrm{0^{3,8}}$ refers to the new linkage between $\ce{C}3$ and $\ce{C}8$ with $0$ carbons between.

Finally, pentacyclo[$4$.$4$.$0$.$\mathrm{0^{2,5}}$.$\mathrm{0^{3,8}}$.$\mathrm{0^{4,7}}$]decane:


Now you might ask: how on earth is that basketane? Let me show you how:

Not convinced? Here is the proof:

$1$ is connected to $2$, $6$, $10$.
$2$ is connected to $1$, $3$, $5$.
$3$ is connected to $2$, $4$, $8$.
$4$ is connected to $3$, $5$, $7$.
$5$ is connected to $2$, $4$, $6$.
$6$ is connected to $1$, $5$, $7$.
$7$ is connected to $4$, $6$, $8$.
$8$ is connected to $3$, $7$, $9$.
$9$ is connected to $8$, $10$.
$10$ is connected to $1$, $9$.

In the above two pictures, you can verify the connectivities, and they match exactly.

Derivation
Firstly, we try to find the largest ring that can contain the most carbon atoms. In this case, it is already indicated above (trace the ring $1$-$2$-$3$-$4$-$5$-$6$-$7$-$8$-$9$-$10$).
Then, you might ask, &quot;How do we judge which numbering is correct?&quot;
Curiously, the 1993 IUPAC Recommendations do not contain the criteria, but the 1979 IUPAC Recommendations do. Rule A-$32.31$, Nomenclature of Organic Chemistry, Sections A, B, C, D, E, F, and H. Pergamon Press, Oxford, 1979. Copyright 1979 IUPAC.:

$32.31$ - When there is a choice, the following criteria are considered in turn until a decision is made:
(a) The main ring shall contain as many carbon atoms as possible, two of which must serve as bridgeheads for the main bridge.
(b) The main bridge shall be as large as possible.
(c) The main ring shall be divided as symmetrically as possible by the main bridge.
(d) The superscripts locating the other bridges shall be as small as possible (in the sense indicated in Rule A-$2.2$).


Let us look at the first rule:

(a) The main ring shall contain as many carbon atoms as possible, two of which must serve as bridgeheads for the main bridge.

This means that we should maximize the sum of the first two numbers and minimize the sum of the other numbers.
Also, this means that we need to choose two carbons which are each connected to three carbon atoms.
For example, this would be wrong:

Incorrect name: pentacyclo[$\color{red}{4}$.$\color{orange}{2}$.$\color{green}{2}$.$\mathrm{0^{2,5}}$.$\mathrm{0^{3,8}}$.$\mathrm{0^{4,7}}$]
This name is incorrect because the first two numbers, $\color{red}{4}$ and $\color{orange}{2}$, add up to $6$, which is smaller than $8$ as derived from the correct name.

Let us look at the second rule:

(b) The main bridge shall be as large as possible.

The &quot;main bridge&quot; refers to the third number, which is to be maximized.
This is not applicable here, as every number beyond the third is zero.

Let us look at the third rule:

(c) The main ring shall be divided as symmetrically as possible by the main bridge.

This means, that although the first number is to be larger than the second, the difference between the first two numbers is to be minimized.
For example, this would be wrong:

Incorrect name: pentacyclo[$\color{red}{6}$.$\color{orange}{2}$.$\color{green}{0}$.$\mathrm{0^{2,7}}$.$\mathrm{0^{3,10}}$.$\mathrm{0^{6,9}}$]
This name is incorrect because the first two numbers, $\color{red}{6}$ and $\color{orange}{2}$, have a difference of $4$, while the difference between the first two numbers in the correct name is $0$.

Let us look at the fourth rule:

(d) The superscripts locating the other bridges shall be as small as possible (in the sense indicated in Rule A-$2.2$).

This is what A-$2.2$ says:

[...] When series of locants containing the same number of terms are compared term by term, that series is &quot;lowest&quot; which contains the lowest number on the occasion of the first difference. [...]

For example, this would be wrong:

Incorrect name: pentacyclo[$\color{red}{4}$.$\color{orange}{4}$.$\color{green}{0}$.$\mathrm{0^{2,9}}$.$\mathrm{0^{5,8}}$.$\mathrm{0^{7,10}}$]
Let us compare the locants in the incorrect version and the locants in the correct version:

Incorrect: $2,9,5,8,7,10$
Correct: $2,5,3,8,4,7$

The first difference is $9$ vs $5$, in which the correct version is smaller.

Conclusion
Therefore, we have arrived at the correct systemic IUPAC name for basketane:

Pentacyclo[$\color{red}{4}$.$\color{orange}{4}$.$\color{green}{0}$.$\color{purple}{\mathrm{0^{2,5}}}$.$\color{blue}{\mathrm{0^{3,8}}}$.$\color{pink}{\mathrm{0^{4,7}}}$]decane.