# A holographic matrix puzzle

Fill the missing square.

It is my own creation. It is not part of a test, so there is no ABCD choice. You must figure out the underlying rule to find the missing pattern.

Before you ask: nothing to do with noughts and crosses (aka tic-tac-toe).

Hint:

The key is in the title.
Don't try to find a row rule or column rule.

Hint:

All x's and all o's are not equal. Maybe this can help.

Hint:

It is marked as "progressive-matrix" because it is the closest. But I don't think "progressive" is the correct term. There is no pregression between the squares.
Also, instead of "holographic" I should better say "auto-holographic".
It should be called an auto-holographic matrix puzzle.

Hint:

Here is an easier version of the problem.

• Just want to make sure. No need for the "knowledge"-tag? Aug 29, 2022 at 10:47
• No. And I guess it is time for another hint. Aug 29, 2022 at 11:18

Is it:

Explanation:
(squares and positions within squares will be referred to as 1-9 via dial pad convention)

The final hint suggests that symbols exist in pairs, as each letter is usually used twice. I noticed in the final hint that on the top row, the h in square 3 was in position 1, and the h in square 1 was in position 3. This gave me the rule:

Symbols are placed in pairs; the position within a square of one symbol should point to the square of the other, and vice versa.

Checking all the pairs in the final hint, and later the original grid, this rule seems to bear out. There is one special case in square 6, where an f is in position 6--in this case, the symbol is its own compliment, and it does not exist as a pair.

Looking at the given eight squares, three have a symbol in position 9. This determines the first eight positions in square 9:

Given just the rule above, there is no way to determine whether the ninth position is empty or contains a self-complimenting symbol. However, looking at the other squares, each contains exactly three symbols. Following this pattern, the final position of the ninth square must be empty.

Aside:

The solution to the easier version given in the last hint spells out win. :D

• Correct. And what makes it holographic? Aug 29, 2022 at 19:55
• @FlorianF Honestly, I'm not sure. I know a hologram involves light interference, or can refer to a projected 3d image, but I'm not sure how that relates to the solution Aug 29, 2022 at 20:30
• Maybe I am abusing the term. In a hologram every part of a hologram contains the image of the whole object. Here each part is reflected in the whole. Aug 29, 2022 at 21:44