With problems like this, I normally try to figure it out based on the set of possible answers. Multiple-choice answers very often contain the correct answer and various permutations of it, so the "consensus" answer is probably the correct one. Looking down the columns, we find the most common letter at each position is:
LRCRPPDQO, which happens to be answer 3. So it's probably 3, but it would be nice to find at least part of an encoding algorithm.
JHXDJQSYA have the same number of letters. So we presumably need only look for algorithms that compute a slot as a function of letters in the other slots, not fancy deletion/expansion stuff.
Next, let's look for repeated letters:
I occurs at positions 4 and 6 (counting from 0) in
STABILIZE but the repeats in
JHXDJQSYA occur at positions 0 and 4. Thus it's not merely a letter-by-letter translation. Furthermore it's not just translation and shuffling, since there are three
RESPONSES but no trio of identical letters in any of the possible answers. Thus, there is presumably an impact of either position or previous letter(s).
For our next hint, let's look at the answers. (2) is just
RESPONSES minus one at each location, and that's clearly not what is happening with
STABILIZE, so we can throw
2 out. Thus,
RESPONSES translates to something that looks like this:
Now we just brute force various operations on the letters and look for something to jump out. The first interesting pattern I found was if you sum the letters (
A == 0):
+--------- (mod 26)
This suggests that whatever factors influence the first four characters, it's the same on the last four (in the same order). So we have a pattern: if you go forward by 5, as many letters as you go up (down) in the original, you go the opposite in the encoded version. For example,
'J' + 'S' - 'L' == 'Q' to get answer character 5 given 0 and the original 0 and 5. We take what we know, and get:
And that leaves only two possible answers: (3) and (4). Do either obey the rule? We have
D; 'R' + 'E' - 'S' is 'D', so
just like we predicted from examining the structure of the answers. Why, exactly? I'm not sure. But we don't really need to know.
Just to check that we really are on the right track, let's sum our proposed answer:
--------- (mod 26)
And then if we look at the difference between these two answers:
C V U G
-B A X E
== == == ==
1 -5 -3 2
And if we look at the relevant pair of letters and sum them, then take the difference between the two words, we get:
'R' + 'N' - 'S' - 'L' = 1
'E' + 'S' - 'T' - 'I' = -5
'S' + 'E' - 'A' - 'Z' = -3
'P' + 'S' - 'B' - 'E' = 28 = 2 (mod 26)
So these values clearly arise from some predictable operations on the letters; the
BAXERBAXE values are not merely random as we can predict the
CVUG pattern from them.
We could guess that there is salt with values (6, 6, -2, -1, 1) and then the encoding formula is:
- First 4 letters: letter at index + letter at (index+5) + salt at index
- Central letter: letter + final salt value (i.e. 1)
- Last 4 letters: coded letter at (index - 5) + letter at (index - 5) + letter at index
This matches everything that is specified, though it's possible that the salt is intended to be the result of some other operation.