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.