# Derive the combination and escape the room!

You awake to find yourself on the floor of what appears to be a cold cell. You don't know how you got there but you're sure no one is coming for you and you need to get out. The stone room has no windows and the only exit is a large oak door.

On the door is some engraved writing and three dials like those of a combination lock (numbered 1 through 50). The door is locked of course. It's got a deadbolt and its hinges are on the other side.

DERIVE THE COMBINATION:

I. THX XXMBER OF XEXTEXX XX THE XXXXXBXT (LXTXX, NOT GXEEK).

II. TXE NXMXXX OF WXXXX TXXXS IN XX XCTXVE

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

NOW
CAN YOU
RESOLVE IT?


It seems parts of the writing have been scratched out. The line below II. is completely scratched out and there is nothing to be read.

How do you get out of here?

Hint:

The third line seems completely unreadable. However, the peculiar arrangement of the final three lines stands out to you. Why were the lines broken like that? It seems like it might mean something...

• Can the third number be the number of Xs on the third line? I'd answer myself but it is impossible to count those on my mobile phone. Way too small. Jun 8, 2016 at 3:28
• You don't see any deliberate X's, only miscellaneous scratches. Jun 8, 2016 at 7:24

I think the combination is:

23 02 30 (or possibly 10 09 47 see below)

First Line:

26 -> THE NUMBER OF LETTERS IN THE ALPHABET. (LATIN, NOT GREEK).

Second Line:

6 -> THE NUMBER OF WHOLE TONES IN AN OCTAVE.

Lastly:

The last three lines start N, C, R and 26 nCr 6 is 230230, which splits nicely into 3 numbers between 1 and 50: 23, 02, 30

Possible Alternative

If 23 is used instead of 26 for the number of letters in the latin alphabet, as suggested in the answer comments, then 23 nCr 6 is 100947, which still works as a combination: 10, 09, 47

• This. The first line and title of the question explicitly tell you to do this. Jun 8, 2016 at 14:06
• Click The door swings open. You are free. Jun 8, 2016 at 15:54
• (You got it with the first one, no alternative needed. Nice job!) Jun 8, 2016 at 15:57
• @Tonkleton Thanks, very much enjoyed the question (and hints), my first puzzling success! :)
– Arth
Jun 8, 2016 at 16:14
• Now, the puzzle is how to calculate 26 C 6 in your head! Jun 8, 2016 at 23:19

Partial Progress:

I: THE NUMBER OF LETTERS IN THE ALPHABET. (LATIN, NOT GREEK). This is 26.
II: THE NUMBER OF WHOLE TONES IN AN OCTAVE. This is 6.

I still can't understand how III is possible...

• WHOLE tones, not HALF tones. Jun 8, 2016 at 0:52
• You feel very confident about the progress you have made, but you know there's more to it. Jun 8, 2016 at 7:22
• I feel like there has to be some sort of unscramble of the X'd out letters. In the first line the X'd letters are: ENULTRSINALPHAEAINR which seems to turn into "RESULT IN A LINEAR " with "PHAN" left over.. so not quite right.. Jun 8, 2016 at 15:39

I: THE NUMBER OF LETTERS IN THE ALPHABET. (LATIN, NOT GREEK).

26

II: THE NUMBER OF WHOLE TONES IN AN OCTAVE

6

III:
NOW (3 letters)
CAN YOU (6 letters)
RESOLVE IT? (9 letters excluding question mark)

12

THE NUMBER OF LETTERS IN THE ALPHABET (LATIN, NOT GREEK) and THE NUMBER OF WHOLE TONES IN AN OCTAVE

giving you the numbers

26 and 6

New possible answer to the third line:

Perhaps "derive the combination" is referring to taking the derivative of something, in which case it would be relevant that all the numbers are scratched out with X's. The third line is 62, which is too long, but if you combine the X's from the first two, there are 33 X's (33X), and if you then derive this combination, you get 33.

• Ah, beat to it! Jun 8, 2016 at 0:51
• Welp, google said 23 and I didn't read further. The romans used 23. I'm assuming you're right. Jun 8, 2016 at 0:53
• You try all the possibilities on the third dial but the door still doesn't open. Jun 8, 2016 at 0:53
• Can we try 23-6 and then bash out the third? Jun 8, 2016 at 0:56
• @SSung2710 The English alphabet uses Latin characters, so I just operated under that assumption. It certainly may be the ancient version Jun 8, 2016 at 0:56

Using the first two lines you get the numbers 26 and 6.

I: THE NUMBER OF LETTERS IN THE ALPHABET. (LATIN, NOT GREEK).

II: THE NUMBER OF WHOLE TONES IN AN OCTAVE

Now for line three:

III: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

I know there are 62 x's in the sequence

So to answer this question I have come up with three solutions that I think are the most prominent.

(26*6) mod 62 = 156 mod 62 = 32

Resulting in:

26, 6, 32

or

Due to the number of X's in line one(19) and line two(16), add those together and subtract them from 62. 62 - (19 + 16) = 62 - 35 = 27

Resulting in:

26, 6, 27

or

Just subtract the previous numbers from 62. 62 - (26 + 6) = 62 - 32 = 30

Resulting in:

26, 6, 30

The combination is:

26, 6, 31 (do this last one twice...)

First:

26 as in I: THE NUMBER OF LETTERS IN THE ALPHABET. (LATIN, NOT GREEK).

Second:

6 as in II: THE NUMBER OF WHOLE TONES IN AN OCTAVE.

And finally:

31 (performed twice) This one is a bit of a stretch. 62 is the number of X's on the wall. Obviously, the input can't be greater than 50.... but 62 is a semiprime number. It is formed as the product of two prime numbers in this case 31 x 2 . So my vote is the third combination is 31, performed twice.