Cyrus' Adventure (Part 1)

Cyrus is on an adventure to the deep unknown forest, in order to get the mysterious treasure box. After walking for a few days, he found out that a strange door was blocking his way, and there weren't any paths that Cyrus could take. Without choice, he could only try to open the door. There were some numbers on the door and strangely, there was a touch-sensing screen near the side of the door. It showed the alphabet, from A-Z. There were also 4 lines on top of the alphabet buttons. Cyrus pressed a random letter, and it appeared on the first line. He quickly found out that it was a password lock. He cleared the letters and started to look at the four numbers. They are 2, 5, 13 and 34. The question comes. What are the 4 letters? What is the password? Can you help Cyrus solve the problem?

p.s. you can go ask your math teacher for hints

• There are 5 lines but we know we only need 4 letters? Is that a mistake? Should it say "What are the four numbers?" perhaps? – Jonathan Allan Feb 2 at 16:44
• @JonathanAllan sorry corrected typo – K Sharing Feb 3 at 3:27

It's apparent that the numbers

are alternate Fibonacci numbers: 0 1 1 2 3 5 8 13 21 34 etc.

I would try

considering the following correspondence: 1/F 1/I 2/B 3/O 5/N 8/A 13/C 21/C 34/I and entering BNCI if we need four letters, and maybe the complement FIOAC if we need five (one per line).

I have to confess that the problem seems somewhat underdetermined: on the face of it (though I may be missing something) there are several different ways to get from the information we have to a sequence of letters.

• +1 for the comment about having multiple ways to get the letters. – ZanyG Feb 2 at 23:57
• You are correct for BNCI! – K Sharing Feb 3 at 3:26

If I were Cyrus I'd give these four a go:

A C H U

...because $$2$$, $$5$$, $$13$$, and $$34$$ are all Fibonacci numbers
($$f(n)=f(n-2)+f(n-1)$$ with $$f(0)=1$$ and $$f(1)=1$$)
i.e.: $$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \cdots$$
and treating A,B,C,... as $$1,2,3,...$$ these are what appear to be the missing numbers as letters.

if we need 5 letters (as there are 5 lines) try

A A C H U

"Gesundheit!"