Patterns - they're everywhere!

You wake up in a room (again) that has two floors, Both have 3 puzzles. In order to escape, You have to find every answer in this order: Red Yellow Green Cyan Navy Pink and combine it together (eg if 75 is Red then 64 is Yellow then the answer is 7564) Most of them are patterns, so get ready!

First Floor

Part 1:

0 -----> 0 -----> 1 -----> 2 -----> 4 -----> 10 -----> 34 -----> Red

Part 2:

1 -----> 4 -----> 7 -----> 11 -----> 15 -----> 19 ----- 24 -----> Yellow

Part 3:

[8, 2, 0, 8, 11], [12, 3, 3, 81, 21] [17, 4, 20, 1024, 1211], Green

Find all the numbers at the next set. Add them all up. If there is a fraction, take the denominator.

Second Floor

Part 4:

0, 1, 5, 16, 41, 92, 189, Cyan

Part 5:

1, 3, 9, 28, 276, 867, Navy

Part 6:

The last puzzle, is math questions!

1. What is the largest possible number that has no "N" in it?
2. If all roman numerals were arranged alphabetically, what would be the last?
3. What is the highest number that has no repeated letters? (English)
4. What is the highest whole number consisting entirely of alternating consonants and vowels?
5. What is the smallest natural number that uses a, e, I, o, and u in order?
6. Add all 3 lettered numbers. What is the result?
7. What is the smallest natural number whose name is spelled with 15 letters?
8. Add all the numbers that only use "E" as a vowel. What is the result?
9. What's the first number that uses "A" as a vowel?
10. Whats the funniest number haha

Question: What is the number for every puzzle and what is the code to escape?

Part 1:

Red = 154. The differences between numbers (except the first zero) form successive factorials:
$$1 - 0 = 1 = 0!$$
$$2 - 1 = 1 = 1!$$
$$4 - 2 = 2 = 2!$$
$$10 - 4 = 6 = 3!$$ $$34 - 10 = 24 = 4!$$
So, the next number is $$34+5!=34+120=154$$.

Part 2 (unsure):

Yellow = possibly 29. The differences between numbers are 2 threes ($$7-4=4-1=3$$), 3 fours ($$19-15=15-11=11-7=4$$), and next should be 4 fives ($$39-34=34-29=29-24=24-19=5$$).

Part 3 (also partly unsure):

The 1st elements of the sets have successive differences: $$12-8=4$$, $$17-12=5$$. So, the next one should be $$17+6=23$$.
The 2nd elements of the sets are consecutive integers: $$2, 3, 4$$. The next should be $$5$$.
The 3rd elements of the set are possibly successive squares multiplied by successive odd numbers (or successive primes): $$0^2\times1=0$$ (or $$0^2\times2=0$$), $$1^2\times3=3$$, $$2^2\times5=20$$. Both pattern yield the same next number $$3^2\times7=63$$.
The 4th elements of the sets are successive integers raised to the power of itself plus one, i.e. $$n^{n+1}$$: $$2^3=8$$, $$3^4=81$$, $$4^5=1024$$. The next should be $$5^6=15625$$.
The 5th elements of the sets are the descriptions of the previous: i.e. 11 has two ones, so the next is 21; 21 has a two and a one, so 1211; 1211 has a two and three ones, so the next will be $$1231$$.
Green = $$23+5+63+15625+1231$$ == 16947.

Part 6:

1. 88 (eighty-eight). 89 is eighty-nine, 90-99 contain the word ninety, and 100 and greater contain hundred (then thousand, million, billion etc). Of course, if we don't use unorthodox names such as googol ($$10^{100}$$).
2. 38 (XXXVIII), since X is the alphabetically last Roman numeral, and V is next-to-last one. (Again, if we don't use unorthodox writing such as XXXXVIIII for 49 instead of XLIX).
3. Possibly 5000 (five thousand). I couldn't get any higher :)
4. 99 (NiNeTy-NiNe, Y is pronounced here as a vowel). Again, 100 or larger numbers are ineligible for nearly the same reason as in Question 1.
5. Possibly 1134 (one thousAnd onE hundred thIrty-fOUr). Again, I couldn't get any lower (assuming that AEIOU are not only vowels in the spelling).
6. 19 = 1 (one) + 2 (two) + 6 (six) + 10 (ten).
7. 103 ("one hundred three" indeed has 15 letters). We must go above 100 since the longest (in the terms of spelling) 2-digit numbers are 73, 77 and 78 (seventy-three, seventy-seven and seventy-eight), which are 12 letters each.
8. 60 = 3 (three) + 7 (seven) + 10 (ten) + 11 (eleven) + 12 (twelve) + 17 (seventeen).
9. 1000 (one thousAnd).
10. Possibly 55 (since the numeral 5 is pronounced as "ha" in Thai (I can be wrong about the exact language), so Thai people actually use "555" or so in SMS writing to express laughter. The question itself has 2 "ha"s, so the answer is probably 55.
So, Pink is possibly 88+38+5000+99+1134+19+103+60+1000+55 = 07596 (we must have a 5-digit number, so adding a leading zero is necessary).

• the 10th number in part 6 does not rely on "haha" Mar 30 '21 at 10:07
• 6.10 could be either rot13(fvkgl-avar be sbhe-uhaqerq-gjragl) Mar 30 '21 at 15:37
• 6. 10 these are my thoughts rot13(bar uhaqerq bar fvapr vg ybbxf yvxr YBY) but samm82 has a good point Apr 30 '21 at 19:35
• In part one you seem to have forgotten the leading $0$. This is not a good match as $\Gamma(s)$ has a pole rather than a zero at $s = 0$. It might be an issue in the design of the puzzle. For part two, there is a unique match on OEIS. Dec 25 '21 at 18:13
• For 6.5 I got rot13(bar gubhfNaq RVtugl sBHe) Dec 25 '21 at 18:28