Code: Codes and Combinations

In one line: with two symbols, every extra position doubles the number of possible codes — 2ⁿ is the most important formula in computing.

Companion to: Charles Petzold, Code (2nd edition), chapter 2.

Counting combinations

• The formula: two symbols in n positions gives 2ⁿ codes.
• Enough for the alphabet: codes of length 1–4 give 30 — enough for 26 letters.

Decoding with a tree

                 (start)
           .  /          \  -
            E              T
         . / \ -        . / \ -
          I   A          N   M
         / \ / \        / \ / \
        S  U R  W      D  K G  O

Read a code left to right: dot = go left, dash = go right, then stop. -.- → T → N → K.

Try it

sum(2**n for n in range(1, 5))   # 30 — Morse codes of length 1 to 4
2**8                             # 256 — values in one byte

[math]::Pow(2, 8)                # 256

🔐 Security and IT connections

• Keyspace is combinations. A password drawn from C characters at length L has C^L possibilities, and each extra character multiplies the work for an attacker.

  Copy

  26**8      # 208,827,064,576 — 8 lowercase letters
  95**12     # ~5.4 × 10^23   — 12 printable ASCII characters

  Length beats complexity: adding characters grows the keyspace far faster than adding symbol types.

• Bits work the same way. A 128-bit key has 2¹²⁸ possibilities — far beyond brute force.

Practice drills

Key takeaways

• Two symbols in n positions give 2ⁿ combinations; each position doubles the total.
• A binary tree decodes a two-symbol code step by step.
• The same maths sizes passwords and keys — length is the lever.