It seems like you're trying to create patterns or relationships between digits of the number 123. Below are a few interesting ways to treat the digits of 123 using puzzle-like operations, inspired by your examples:

Possible Puzzles with Digits of 123:

1. Sum of consecutive digits:

   • $1 + 2 = 3$
   • $2 + 3 = 5$

2. Pair product followed by addition:

   • $1 \times 2 + 3 = 5$

3. Concatenate and add in stages:

   • $1 \rightarrow 12$, $12 + 3 = 15$

4. Difference between consecutive digits:

   • $2 - 1 = 1$
   • $3 - 2 = 1$

5. Factorial manipulation:

   • $1! + 2! + 3! = 1 + 2 + 6 = 9$

These are just creative ways to play with the digits mathematically. Do you have any particular puzzle rule you want to explore further?

---

Would you like more patterns or solutions?
Here are 5 questions to expand on this:

1. Can we use subtraction, division, or other operations on these digits?
2. What happens if we reverse the digits (e.g., 321)? Any new patterns?
3. Can we treat the digits as part of a sequence and explore sums?
4. Are there ways to use exponents or roots with these digits?
5. Can we treat 123 as a whole and explore number properties like prime factors?

Tip: Playing with digit sums can help find quick patterns for divisibility rules, like checking if a number is divisible by 3 or 9.