📊 Base‑b Recursive Listing Method

🧠 Concept

• Idea: Because of positional representation, each digit “switches” every power of the base according to how far it is from the smallest unit.
• Implication: For base $b$, the $2^\text{nd}$ row (from the top) changes every $b$ entries, the $3^\text{rd}$ row every $b^2$ entries, etc.
• Result: To list all combinations up to some digit depth, draw blocks whose sizes are powers of $b$ and recursively split each block into $b$ parts.

🔢 Example: base $3$, up to the $3^\text{rd}$ digit (MSD → LSD)

• First row (MSD):

  • Block size: $3^2=9$
  • Action: Write each digit $0,1,2$ exactly $9$ times in sequence.

• Second row (middle digit):

  • Inside each MSD block of $9$: split into $3$ parts of size $3$ and label them $0,1,2$.

• Third row (LSD):

  • Inside each size‑$3$ block: split into $3$ single entries and label them $0,1,2$.

This produces every ternary triple from $000$ to $222$ in lexicographic order.

🧭 General method (base‑agnostic)

$b$ for base

• At the Most Significant Digit (leftmost digit),
  • List $b^{(\text{positions from the first digit})}$
    • Incrementally from $0$ to $(\text{base}-1)$,
  • Then split each part into $b$ parts recursively until you reach the LSD.

Recursively here means you do the same splitting for each sub-part