Shifted accordingly

Question

Replace the question marks with the correct letters and then tell me what word I've hidden below.

---

                            (5)          
                   -        (4)            
                 .--        (5)           
              ...---        (8)         
            ....----        (13)     
          .....-----        (8)       
        ......------        (7)
             

             


      P   ?   I   N   ?   ?   J   O   F   C   

      P   I   N   ?   W   U   L   D ?(+1) J   

      O   H   U   Y   E   H   V   M   P   ?   

      X   I   V   ?   P   U   J   S   ?   F   

      L   T   S   N   K   D   G   M   U   T
             

  
          
             

Answer 1

The hidden word is:

GEOMETRY

How did I work this out? First I focussed on:

(i) the word 'shift' in the title, suggesting the involvement of a Caesar shift cipher;

(ii) the fact that the sum of all the numbers in the parentheses is 50 - the same as the total number of entries in the grid beneath.

This made me try...

...splitting the grid into blocks of the lengths indicated in parentheses, giving me:

(5) P?IN?
(4) ?JOF
(5) CPIN?
(8) WULD?JOH (this question mark is the one with '+1' linked to it)
(13) UYEHVMP?XIV?P
(8) UJS?FLTS
(7) NKDGMUT

At this point I noticed that the first 'word' could spell 'POINT' if the question marks represented a missing 'O' and 'T'. This word has had no Caesar shift applied to it at all, and there are no dashes to the left of it. Could the number of dashes represent the amount of places by which to shift the letters in the alphabet for each word? I tried this and got some promising results by shifting each letter backwards by the amount in question:

(5) P?IN? (0) P?IN?
(4) ?JOF (-1) ?INE
(5) CPIN? (-2) ANGL?
(8) WULD?JOH (-3) TRIA?GLE
(13) UYEHVMP?XIV?P (-4) QUADRIL?TER?L
(8) UJS?FLTS (-5) PEN?AGON
(7) NKDGMUT (-6) HEXAGON

Wait, that last one absolutely spells HEXAGON! And it's clear from the others that there are particular letters that can be inserted to make the set of words: POINT, LINE, ANGLE, TRIANGLE, QUADRILATERAL, PENTAGON and HEXAGON - a sequence of shapes of increasing complexity.

Oh, so now we see what the dots and dashes to the left represent - each dash is a line in one of these shapes, and each dot is a vertex (e.g. 6 vertices and 6 lines in a hexagon, but just 1 vertex and 2 lines in an angle). And the name of each of these shapes has been shifted accordingly (as per the title) based on the number of lines they are made up of.

Great, so this is pulling together. So how to find the final hidden word?

How about working out what letters would have appeared in the original grid before all the Caesar shifts? This gives us:

(5) P?IN? (0) P?IN? missing O and T (stay the same: O, T)
(4) ?JOF (-1) ?INE missing L (+1 gives M)
(5) CPIN? (-2) ANGL? missing E (+2 gives G)
(8) WULD?JOH (-3) TRIA?GLE missing N (+3 gives Q, then an additional +1 as labelled in the original grid gives R)
(13) UYEHVMP?XIV?P (-4) QUADRIL?TER?L missing A and A (+4 gives E, E)
(8) UJS?FLTS (-5) PEN?AGON missing T (+5 gives Y)
(7) NKDGMUT (-6) HEXAGON (no missing letters)

i.e. OTMGREEY, which anagrams to GEOMETRY!