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Can you complete these groups or identify the pattern behind?

Example 1:

Group A : {A,C,E,G,I,....}
Group B : {B,....}

Example 2:

Group A : {C,F,I,L,...}
Group B : {E,J,O...}
Group C : {A,B,D,G,H....}

Example 3:

Group A : {A,B,C,D,E,F,G,H}
Group B : {H,I,J,K,L,M,N}
Group C : {N,O,P,Q,R,S,T,U}
Group D : {U,V,W,X,Y,Z,A}

Answers:

Example #1:

Sorting the letters in a 2x13 grid, each row is the same as each group

Example #2:

Counting from A=1, B=2, etc, Group A includes the multiples of 3, Group B includes the multiples of 5 and Group C the remaining letters

Example #3:

Put the letters in the perimeter of a 7x8 square, beginning in the top-left corner and going clockwise. Each side of the square is represented by each group

"That was easy!" you could be thinking... But, that was the warm-up stage!


For the real question, I have found another way to sort the letters and create some groups. Could you find the groups and also identify the pattern I followed to create them?

Description of the groups:

  • There are 3 main groups, let's call them $A, B$ and $C$. They don't share elements between them, this is: $A \bigcap B = A \bigcap C = B \bigcap C = 0 $

  • The relation between the 3 main groups and the pattern, is that there are 6 additional subgroups $SG_1$ to $SG_6$ can be formed with elements from $A,B$ and $C$. These new subgroups represent more precisely the underlying pattern

Tips:

  • Many words use letters from all 3 main groups, but I will give you some that only use letters from 2 of them:

    • Words with letters from just 2 groups: manager, watch, needle, pattern, found, belly, when, main.

    • Words with letters from all 3 groups: transformed, dreams, trouble, morning, element, puzzle, riddle.

  • The secondary groups can be made in this way: Take 4 letters of one of the main groups, take additional 4 from the next main group and finally take one more from the third main group, thus, each subgroup contains 9 letters and some subgroups share elements.

Based on this information, Can you...

Identify the 3 main groups?: A, B and C.

Identify the 6 secondary groups?: SG1 to SG6.

Identify what the main and secondary groups represent in the pattern?

Tell what is the pattern based on?

The answer is compound of four letters:

____,____,____,____


EDIT: As stated in the comments, it could not be possible to solve the whole question without some additional information. So using hints 1 and 2A will help in solving the first part (main groups)


Any feedback on this post is appreciated (Difficulty, explanation, wording, suggestion...) as well as partial answers, solving procedure, etc

Hints policy: A hint will be given every 50 views for the first three hints. After that, 1 hint every 100 views or 15 votes.


Hint #1 (50 views): (More examples words)

More examples that might help you:
- Words with letters from just 2 groups: that,root,week,tree,seeded,bath,day,new,key,yaw
- Words with letters from all 3 groups: than,more,question,back,weak,referrer

Hint #2 (100 views): (More example words (A) and more details about groups and subgroups (B))

  • Part A:

- First of all, additional words for making the grouping easier:
- Words with letters from just 2 groups: quiz, jacuzzi, two, four, six, june.
- Words with letters from all 3 groups (but this are evenly distributed on the groups, this is, the same number of letter from each group): dreams, answer, december, language, oxygen, neutron, witness,frozen
- The size of the main groups are 6, 8 and 12.

  • Part B:

- Now, more details about the subgroups:
- As said before, each subgroup contains 9 elements. And for each subgroup exists a set of 3 subgroups $SG_A$,$SG_B$ and $SG_C$ that their intersection contains one element: $$ SG_A \bigcap SG_B \bigcap SG_C = 1 $$ - Additionally, each subgroup shares more than one element with the other subgroups except one, which has no common elements: $$ SG_A \bigcup SG_X > 1 \text{ }\forall\text{ } X \text{ in \{B,C,D,E\}} $$ $$ SG_A \bigcup SG_F = 0 $$
- And here are the subgroups partially filled: $$ SG_1:\{A,I,D,...\} $$ $$ SG_2:\{D,U,M,...\} $$ $$ SG_3:\{C,R,L,...\} $$ $$ SG_4:\{R,Z,U,...\} $$ $$ SG_5:\{F,W,N,...\} $$ $$ SG_6:\{I,X,Q,...\} $$

Hint #3 (150 views): (Additional subgroups details (A) and relation between groups and subgroups (B))

  • Part A:

Think about how could you sort the 26 letters in a geometric shape using a single wildcard *, and then, from that shape generate 6 groups of 9 letters each. For example, if you use a 3 by 9 grid and put a * in the middle, you form a rectangle.

ABCDEFGHI
JKLM*NOPQ
RSTUVWXYZ

  • Part B:


- Each letter in the main group $A$ is only present in one subgroup
- Each letter in the main group $B$ is only present in three subgroups
- Each letter in the main group $C$ is only present in two subgroup

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  • 1
    $\begingroup$ So I've managed to classify more than half the letters (fourteen) into three groups, but I think there's not yet enough information given to do the rest. $\endgroup$ – Rand al'Thor Sep 3 at 17:45
  • $\begingroup$ @Randal'Thor did you use the hint #1?... There are some pieces of information I left out of the question for next hints in case someone could infer the idea without them. $\endgroup$ – gustavovelascoh Sep 3 at 21:05
  • $\begingroup$ Yep. Without hint #1 I would've made way less progress. $\endgroup$ – Rand al'Thor Sep 3 at 21:21
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The three groups are:

  1. E, K, M, N, P, V

  2. A, C, G, I, R, T, X, Z

  3. B, D, F, H, J, L, O, Q, S, U, W, Y

Or in numbers:

  1. 5, 11, 13, 14, 16, 22

  2. 1, 3, 7, 9, 18, 20, 23, 26

  3. 2, 4, 6, 8, 10, 12, 15, 17, 19, 21, 23, 25

It's notable that all three groups are

symmetric: every letter is in the same group as its Atbashed inverse.

So far this has been solved by pure logic, using the information given (including in hints #1 and #2) about 2-group words, 3-group words, and symmetric-3-group words. The next step should be a flash of inspiration to understand what these groups and subgroups are.

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  • $\begingroup$ That's awesome progress! However, the pattern you noticed (third hidden block) could be seen as a coincidence or a consequence of the original sorting of the letters into the 6 subgroups. Nice work! $\endgroup$ – gustavovelascoh Sep 4 at 11:24
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Based on Rand al'Thor's sorting of the letters I think I've found the pattern

Arrange the letters into a $3 \times 3 \times 3$ cube with a * in the middle as follows.


Layer 1         Layer 2        Layer 3 A B C          J K L           R S T D E F          M * N           U V W G H I          O P Q           X Y Z
Then, the three main groups are the

Corner pieces
{A, C, G, I, R, T, X, Z}

Edge pieces
{B, D, F, H, J, L, O, Q, S, U, W, Y}

The central pieces on each face
{E, K, M, N, P, V}

And the six subgroups are the letters on each of the six faces.

Subgroup 1
{A, B, C, D, E, F, G, H, I}
Subgroup 2
{A, D, G, J, M, O, R, U, X}
Subgroup 3
{A, B, C, J, K, L, R, S, T}
Subgroup 4
{R, S, T, U, V, W, X, Y, Z}
Subgroup 5
{C, F, I, L, N, Q, T, W, Z}
Subgroup 6
{G, H, I, O, P, Q, X, Y, Z}

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  • $\begingroup$ rot13(jryy qbar! Qvq lbh hfr gur 3eq uvag?) $\endgroup$ – gustavovelascoh Sep 5 at 12:26
  • $\begingroup$ @gustavovelascoh Yes, I did. I hadn't realised how recently you put that up. $\endgroup$ – hexomino Sep 5 at 13:31

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