Cryptarithms

Summary: Mathematical puzzles where letters represent unique digits (0-9). Solve the equation to find the letter-to-digit mapping, then decode a word or phrase.

Example:

  SEND
+ MORE
------
 MONEY

Solution Process:

1. M must be 1 (carry from thousands place)
2. S must be 8 or 9 (to generate carry)
3. From column analysis:
   • S = 9, E = 5, N = 6, D = 7
   • M = 1, O = 0, R = 8, Y = 2
4. Verification: 9567 + 1085 = 10652 [correct]

Decoding: If the puzzle asks to decode "MONEY" = 10652

Another Example:

  CAT
+ DOG
-----
 BIRD

Solution: C=7, A=2, T=9, D=6, O=5, G=3, B=8, I=1, R=4 (729 + 653 = 1382)

Multiplication Example:

   AB
x  C
-----
  DEF

Solution Process:

1. AB is a 2-digit number, C is single digit, result is 3-digit
2. So AB must be at least 10, and C at least 2 (10x2=20)
3. Maximum: 99x9=891, so D can be 1-8
4. Testing systematically:
   • If C=3, AB could be 34-99 to give 3-digit result
   • Try A=2, B=5, C=4: 25x4=100 (D=1, E=0, F=0)
   • But E=F=0 means duplicate!
   • Try A=1, B=3, C=7: 13x7=91 (only 2 digits)
   • Try A=2, B=4, C=3: 24x3=72 (only 2 digits)
   • Try A=3, B=9, C=2: 39x2=78 (only 2 digits)
   • Try A=3, B=6, C=4: 36x4=144 (D=1, E=4, F=4 duplicate!)
   • Try A=2, B=7, C=4: 27x4=108 (D=1, E=0, F=8) [valid!]
5. Solution: A=2, B=7, C=4, D=1, E=0, F=8
6. Verification: 27 x 4 = 108 [correct]

Division Example:

    HI
   ----
FG | JKL

Find the values where a 2-digit number divided by another 2-digit gives result HI.

Example solution: 12 | 84 = 7 (F=1, G=2, J=8, K=4, H=0, I=7 -> but H can't be 0 in quotient) Better: 156 / 13 = 12 (J=1, K=5, L=6, F=1, G=3, H=1, I=2 - duplicates!)

Valid solution: 84 / 12 = 7 (if using single digit result)