I might be dumb but at least I have introspection.

This is how my brain does multiplication: I categorise each fact based on the level of cognitive effort they require, ranging from intuitive to "yikes".

• Intuitive: Immediate and automatic, without needing further thought.
• Stored: Retained in memory, feeling slightly less reliable than “intuitive“, but has some sort of intuition in it. If challenged, I would not reconsider.
• Memorised: Also retained, but "less securely". It has to be remembered without intuition. If challenged, I might briefly reconsider. One could say it is in the cache not in the RAM.
• Addition or subtraction or division: These operations became intuitive after a “step“.

2 X 2 up to 2 X 9 = intuitive

3 X 1 up to 3 X 3 = stored
3 X 4 = 8 + 3
3 X 5 = intuitive
3 X 6 = 15 + 3
3 X 7 = 14 + 7
3 X 8 = 16 + 8
3 X 9 = 30 - 3

4 X 3 = 8 + 4
4 X 5 = intuitive
4 X 6 = 12 + 12
4 X 7 = 14 X 2 = 20 + 8
4 X 8 = 16 X 2 = 20 + 12
4 X 9 = 40 - 4

• Note: There seems to be something about 4’s and 8’s that causes difficulty.

5 X 5 up to 5 X 6 up to 5 X 9 = (N/2)10

• Note:  No cognitive effort with 5's.

6 X 6 = stored
6 X 7 = memorised 
6 X 8 = 6 X 7 + 6 = 42 + 6
6 X 9 = 60 - 6

• Note: 6 X 8 causes significantly more cognitive trouble than every other operation. For some odd reason it is even worse than "memorised".  It starts with "It is not 8 X 8". I suspect that 8 as "first" multiplicator is statistically seldom (since any number lower than 8 becomes the "first" multiplicator instead. From a cognitive perspective 5 X 8 isn't the same as 8 X 5) and the brain doesn't develop strategies.

7 X 7 = memorised
7 X 8 = (7 X 7) + 7 = 57 - 1
7 X 9 = 70 - 1

8 X 8 = memorised
8 X 9 = 80 - 8

9 X 9 = 90 - 9

• Note:  Almost no cognitive effort with 9's.

10 X 1 up to 10 X N = intuitive

11 X 1 up to 11 X 9 = (10N) + N
11 X 11 = 10 X 11 + 11 = 110 + 11

12 X 11 = 12 X 10 + 11 =  120 + 11
12 X 12 = memorised

From here one type of cognition is not enough, conscious effort has to be made.

How is this different from your cognitive process? Do you think LLMs might develop something like this in their perceptrons between their transformer layers?