# What's the pattern?

I went to a library and found a book which contained puzzles. I found this puzzle, and decoded it. I showed it to my friends, and they couldn't figure out. I then posted it on this website...

Find the pattern in these numbers:

12, 18, 24, 72, 16, 8, 4, 24, 30, 36, 42, 126, 28, 14, 7, 5040


Note: for stuff such as +5 for 3, then +10 for 1, repeat is allowed

Hint:

The first 3 are adding the number which is the square root of 9 + the square root of 9.

• If you're allowing answers like "+5 for 3, then +10 for 1" where the "pattern" changes every now and then, what advantage do such answers have over just, y'know, listing the numbers? – Gareth McCaughan Sep 16 '16 at 16:08
• Sorry, just was being unclear. – Star OS Sep 16 '16 at 16:16
• I agree with Sconibulus's analysis, but personally I wouldn't call this "a valid pattern". – Gareth McCaughan Sep 16 '16 at 16:29
• I hope Matsmath's advice isn't intended entirely seriously. I don't think any puzzle is much improved by adding "irrelevant, random noise" in that fashion. – Gareth McCaughan Sep 16 '16 at 17:08
• @Matsmath: "Difficult" is not the same thing as "good". Irrelevant information makes puzzles worse - ideally, a puzzle will use all of its information. – Deusovi Sep 16 '16 at 17:40

It looks like it's something resembling

12,18,24 = +6
24,72 = x3
72,16 = *2/9 16,8,4 = /2
4,24 = !
24,30,36,42 = +6
42,126 = *3
126,28 = *2/9
28,14,7 = /2
7,5040 = !

So the final sequence is probably:

12, 18, 24, 72, 16, 8, 4, 24, 30, 36, 42, 126, 28, 14, 7, 5040, 5046, 5052, 5058, 5064, 15192, 3376, 1688, 844, 844! (which I'm not even going to try to do)...

• You were pretty close. It does have a factorial, division and multiplication, and it's clear it has addition. The values are a bit off, though. – Star OS Sep 16 '16 at 16:32
• It looks to me as if Sconibulus's description does match what your sequence does. Are you saying you have another sequence of operations in mind that's "better" somehow? Is it better because it's simpler, or what? – Gareth McCaughan Sep 16 '16 at 17:10
• No, just that this one is very close but not the actual answer. – Star OS Sep 16 '16 at 17:39
• Sconibulus' sequence seems to be perfect (+1), maybe except for the fact that it has 2 cycle that do not have a perfect match. In the firs cycle + 6 is made two times, three times in the second. So he can transform 24 into 30 for example multiplying by 5/4 or adding the sum of its digits (2+4 = 6). So the sequence repeats itself 2 times but there is no number after 5040 to check the last operation. +6, +6, x3 ,x3, x2/9, /2, /2, !, x5/4, and again. – marcoresk Sep 16 '16 at 22:04