The thirteen pairs of letters given by an evil troll are -
A-V
B-W
C-Q
D-M
E-K
F-U
G-N
H-P
I-O
J-R
L-X
S-T
Y-Z
And 5 short words given by troll are - FACE, QUEST, QUICK, SWITCH, and WORLD.
As described in the given details, we'll refer letter from password as PASSWORD letter & other as OTHER letter.
As per troll, those short words are having same number of PASSWORD letters.
STEPS :
1] Both S & T are appearing in the pair with each other. Hence, either S or T must be a PASSWORD letter but not both. Since, both letters are appearing in short word QUEST, that is QUEST having at least 1 PASSWORD letter for sure hence, all 5 must have at least 1 PASSWORD letter.
2] Suppose every short word has 1 PASSWORD letter. With S or T as 1 PASSWORD letter from QUEST, other letters Q, U, E can't be PASSWORD letters.
If Q, U, E are not PASSWORD letters then C (from C-Q pair), F (from F-U pair) and K (from E-K pair) must be PASSWORD letters.
In that case, FACE will have 2 PASSWORD letters viz. C & E which goes against our assumption of having exactly 1 PASSWORD letter in each short word.
3] Let's
assume along with S or T the second PASSWORD letter is E i.e each short word has 2 PASSWORD letters. Again, Q, U
can't be PASSWORD letters but C (from C-Q pair) & F (from F-U pair) must be. Still then FACE will
have 3 PASSWORD letters which goes against our assumption of exactly 2 PASSWORD letter in each short word.
4] Now, let's assume along with S or T the second PASSWORD letter is U. Again, Q, E can't be PASSWORD letters but C (from C-Q pair) & K (from E-K pair) must be. Still then QUICK will have 3 PASSWORD letters which goes against our assumption of exactly 2 PASSWORD letter in each short word.
5] Let's assume there are 4 PASSWORD letters in each short word. So apart from S or T, the letters Q, U, E of short word QUEST must be PASSWORD letters.
If Q, U, E are PASSWORD letters then C (from C-Q pair), F (from F-U pair) and K (from E-K pair) must NOT be the PASSWORD letters.
In the case, the short word FACE will have maximum only 2 PASSWORD letters (not sure about A from A-V pair) which again goes against our assumption of exactly 4 PASSWORD letter in each short word.
6] Hence, each short word must be having 3 PASSWORD letters.
If Q, E are the PASSWORD letters with S or T in QUEST, then C & K can't be PASSWORD letters. With that, Q, U, I will be 3 PASSWORD letters in QUICK. And if U too is the PASSWORD letter then QUEST will have 4 PASSWORD letters.
If Q, U are the PASSWORD letters with S or T in QUEST, then C & F can't be PASSWORD letters. With that, FACE can have maximum of only 1 PASSWORD letter.
7] Hence, U & E must be other 2 PASSWORD letters apart from S or T in short word QUEST. So Q must not be the PASSWORD letter but C must be. Also, F and K can't be the PASSWORD letters. Hence, FACE will have E, C and A as PASSWORD letters.
If A is PASSWORD letter then V (from A-V pair) can't be the PASSWORD letter.
8] Next, from QUICK we will have, C, U and obviously I as 3 PASSWORD letters after Q, K are ruled out. If I is PASSWORD letter then O (from I-O pair) can't be the PASSWORD letter.
9] Just like QUEST, SWITCH too have either S or T as PASSWORD letter. Moreover, it has I & C as PASSWORD letters. Hence, H & W must not be the PASSWORD letters.
10] So, if W & O are not the PASSWORD letters then other 3 letters of WORLD i.e. R, L, D must be PASSWORD letters. With that M (from D-M pair), J (from J-R pair) and X (from L-X pair) are ruled out.
11] So far we have got -
PASSWORD letters - U, E, C, A, I, R, L, D, Either S or T.
OTHER letters - Q, F, K, V, O, H, W, M, J, X
12] Arranging every OTHER letter in alphabetical order & writing down corresponding PASSWORD letter below it -
OTHER : F H J K M O Q V W X
PASS. : U P R E D I C A B L
13] Now, S-T, G-N, Y-Z are the only 3 pairs left. And correct placement for these pairs must be like.
OTHER : F G H J K M O Q S V W X Z
PASS. : U N P R E D I C T A B L Y
CONCLUSION :
The PASSWORD that an evil troll has set must be UNPREDICTABLY.
