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This puzzle is inspired by JLee's What is a Word/Phrase™ series and the subsequent "Number" variants. (Actually, I'd originally tried to create a more original puzzle using the same idea, but it turned into another one of these. Sigh. Maybe the types of numbers given lend this somewhat more novelty.)

If a positive real number obeys a special rule, I call it a Good Pasta Number™. Here are some examples:

A good, solid pasta! Yeah! You are a horrible cook
$6\sqrt{6}$ $3\sqrt{5}$
$6$ $8$
$3\sqrt{6}$ $6\sqrt{3}$
$\frac{3\sqrt{2}+\sqrt{6}}{2}$ $\frac{3\sqrt{6}+\sqrt{2}}{2}$
$3\sqrt{50-22\sqrt{5}}$ $3\sqrt{50+22\sqrt{5}}$
$9\sqrt{3}-3\sqrt{15}$ $18\sqrt{3}-6\sqrt{15}$
$\frac{42\sqrt{6}}{23}$ $\frac{56\sqrt{6}}{19}$
$\frac{15\sqrt{6}-18\sqrt{3}+135\sqrt{2}-162}{14}$ $\frac{15\sqrt{30}-18\sqrt{15}+135\sqrt{10}+30\sqrt{6}-162\sqrt{5}-36\sqrt{3}+270\sqrt{2}-384}{14}$

There are many more Good Pasta Numbers™ not shown above, but only a finite amount.

Which numbers make a good pasta™?

Note:

There's nothing hidden in the flavortext other than a very subtle connection (not really a clue); this really is just a normal "What is a _______" puzzle. I like making my puzzles fun to read. Also, the property is specific to the numbers themselves, and not any specific representation of the aforementioned numbers.

Follow-up: It has been 2 years, and I cannot for the life of me remember the rule here. However, I can remember that it has something to do with

Regular polyhedra, and their surface areas and volumes.

Sorry about that, but maybe someone more insightful than me can help jog my memory.

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  • $\begingroup$ In the fifth row, the two numbers are Galois conjugate. Therefore I suppose that all your numbers are considered as real numbers and the square roots are positive ones. It might be useful to clarify this. $\endgroup$
    – WhatsUp
    Commented Aug 5, 2022 at 14:21
  • $\begingroup$ Uh, I'm not exactly sure what needs clarification. I already said "if a positive real number obeys a special rule," and square roots of positive numbers are positive by definition in most contexts. $\endgroup$
    – Auride
    Commented Aug 5, 2022 at 15:23
  • $\begingroup$ Sorry, you are right, I missed the "positive real number" part. $\endgroup$
    – WhatsUp
    Commented Aug 5, 2022 at 16:41
  • $\begingroup$ Can you just take time to rewrite this puzzle? $\endgroup$
    – 1357924680a
    Commented Jun 20 at 11:40
  • $\begingroup$ Well, rewriting a puzzle would be difficult if you didn't actually know the solution to the puzzle... $\endgroup$
    – Auride
    Commented Jun 20 at 19:01

1 Answer 1

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A good pasta number seems to be one that is the

ratio of the coefficients of a regular polyhedron's surface area to its volume.

Alternatively,

A * e / V, where A is the surface area, V is the volume, and e is the edge length.

For example,

6 is a good pasta number as $A_{cube} = 6a^2$ and $V_{cube} = a^3$, and $6 = \frac{6}{1}$.

Similarly,

$A_{tetra} = a^2\sqrt{3}$ and $V_{tetra} = \frac{a^3}{6\sqrt{2}}$, so $\frac{A}{V}*a = 6\sqrt{6}$.

Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron are elements 1, 2, 3, 5, and 6 of the list (respectively) - I'm sure with more digging you could find the other elements. I'm not entirely sure what the relevance of the negative examples are, but they might just be similar numbers with no other meaning.

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  • $\begingroup$ Good enough. I'm sure the other numbers correspond specifically to Archimedean solids; which ones, I don't recall. $\endgroup$
    – Auride
    Commented Jun 20 at 19:08

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