CRYPTARITHM

What is a Cryptarithm?

• Letters replace digits in an arithmetic problem (addition, subtraction, multiplication)
• Each letter consistently represents the same digit (0–9)
• Each digit is represented by exactly one letter
• Leading letters are never zero
• Appears as Math (M) at all levels

Classic Example — SEND + MORE = MONEY

Constraints to Use Immediately

• Leading digits ≠ 0 — the leftmost letter of any number is not zero
• If the sum has more digits than either addend, the leading digit of the sum = 1 (it's a carry)
• In SEND + MORE = MONEY → M = 1 (carry from addition)
• From M=1 → O is constrained by the thousands column

Solving Strategy

• 1. Write out columns right-to-left
• 2. Identify forced values first (leading digit of sum = 1)
• 3. Work column by column, tracking carry
• 4. Use modular arithmetic: if D + E ≡ Y (mod 10), only certain (D, E) pairs work
• 5. Try and eliminate — use a tracking grid of remaining digits

Worked Mini-Example

• Units: B + A = C (no carry) or B + A = 10 + C (carry 1)
• Tens: A + B + carry = C
• Both columns give C → A + B = C (same sum both ways means carry = 0 in units)
• CC is a 2-digit repdigit: try C=9, A+B=9, e.g. A=4, B=5 → 45+54=99 ✓

Practice Problem

• Hint: G = 1 (carry). Start there.

Key Takeaways

• Each letter = unique digit 0–9; leading letters ≠ 0
• Carry in each column is 0 or 1
• Start with forced values (carries, leading-digit-of-sum = 1)
• Appears as Math (M) — this is the only M-type cipher this year
• Next class: Porta Cipher — a keyed polyalphabetic cipher