For 3X, it's about 114, and for 4X, 144, which naturally, is twice 72.
These are close, back of napkin, results. With smart phone apps offering scientific calculators, you should get comfortable just taking the nth root of a number for a more precise answer.
Update in response to Brick's comment.
The rule of 72 says that (n)(y)=72 to double your money. It answers both questions, how much time do I need, given a rate, and how much return do I need, given a time?
Logic tells me that if 72 is the number to double, 144 is the 4X. But I'm a math guy, and my logic might not be logical to OP. So -
Let's take the 20th root of 4.
This is the key to use. 4, (hit key) 20, equals. The result is 1.07177 or 7.177%. And this is the precise rate you'd need to quadruple your money in 20 years) Now (n)(y)= 20* 7.177 = 143.55 which rounds to 144. "Rule of 144" to quadruple your money.
This now answers OP's question, "How to derive a Rule of X" for a return other than doubling.
One more example? I want 10X my money. Of course I need the initial guess to enter one calculation. People like 8%, in general. It's a bit below the 10% long term S&P return, and a good round number. The Rule of 72 says 9 years to double, so, 18 years is 4X, and 36 years is 8X. For my initial calculation, I'll use 40 years. The 40th root of 10. I get 5.925% (Again the precise rate that gives 10 fold over 40 years) and multiplying this by 40, I get a "Rule of 237" which I'm tempted to round to 240.
At 6%, 237/6= 39.5 yrs, 1.06^39.5 = 9.99
At 6%, 240/6= 40.0 yrs, 1.06^40.0 = 10.29
You can see that you lose some accuracy for the sake of a number that's easier to remember, and manipulate. 72 to double is pretty darn accurate, so I'll stick with "Rule of 237" to get 10X my money.
To close, the purpose of these rules is to create the tool that lets you perform some otherwise tough calculations away from any electronic device. Of course I know how to use logs, and in real life I'm paid to explain them to students who are typically glad when that chapter is over. I've shown above how the "Rule of X" can be formulated with a power/root key, which, for most people, is simpler. Ironically, log calculations as @jkuz offered, force a continuous compounding which may not be desired at all. It would give a result of 230 for my 10X return example, and the following (using the first equation he offered) -
At 6%, 230/6= 38.3 yrs, 1.06^38.3 = 9.31
which is further away from the desired 10X than my 237 or rounded 240.