Digits in Disguise (Cryptarithms)

What are Cryptarithms?

Cryptarithms are puzzles where digits are replaced by letters. Rules:

1. Each letter represents a unique digit ($0-9$).
2. The first digit of a number cannot be 0.
3. Standard arithmetic rules apply.

Example 1: Addition

Solution:

1. Units Column: $1 + B$ ends in $0$.
   • $1 + B = 10 \implies B = 9$.
   • Carry over 1.
2. Tens Column: $1 (\text{carry}) + A + 1 = B$.
   • Since $B = 9$, $2 + A = 9 \implies A = 7$.
3. Check: $71 + 19 = 90$. Correct.

Example 2: Multiplication

Analysis:

1. Units Digit: $B \times 5$ ends in $B$.
   • Possible values for $B$: $0$ or $5$.
2. If B = 0: $A0 \times 5 = CA0$.
   • $5 \times A$ ends in $A$. Possible $A$: $5$ (since $5 \times 5 = 25$).
   • If $A=5, B=0$: $50 \times 5 = 250$.
   • $C=2, A=5, B=0$. Fits the pattern $CAB$.
3. If B = 5: $A5 \times 5 = CA5$.
   • $5 \times 5 = 25$ (Carry 2).
   • $(5 \times A) + 2$ ends in $A$.
   • Try $A=2$: $(5 \times 2) + 2 = 12$. Ends in 2.
   • If $A=2, B=5$: $25 \times 5 = 125$.
   • $C=1, A=2, B=5$. Fits.

Answer: Two solutions exist ($50 \times 5 = 250$ or $25 \times 5 = 125$).