Can you find the pattern behind the proposed letter grouping?

Question

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!

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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:

____,____,____,____

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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)

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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.

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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

Answer 1

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.

Answer 2

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}