A cryptarithmetic puzzle (also known as an alphametic) is a mathematical brainteaser where digits are replaced with letters: $$\begin{array}{r@{\quad}c@{\quad}c@{\quad}c} & \text{T} & \text{W} & \text{O} \\ + & \text{T} & \text{W} & \text{O} \\ \hline \text{F} & \text{O} & \text{U} & \text{R} \end{array}$$

The Rules:

1. Each distinct letter represents a unique decimal digit from $0$ to $9$.
2. Leading digits cannot be zero ($\text{T} \neq 0$ and $\text{F} \neq 0$).
3. The addition must be mathematically correct!

Mathematical Deductions:

• The Thousands Carry: Adding two 3-digit numbers can produce at most $999 + 999 = 1998$. Therefore, the carry into the thousands column $\text{F}$ must be $1$!
• The Ones Column: $\text{O} + \text{O} = \text{R}$ (plus any carry to the tens). Since $\text{O} + \text{O}$ is even, $\text{R}$ must be an even digit ($0, 2, 4, 6, 8$).
• The Hundreds Column: $\text{T} + \text{T}$ produced a carry into $\text{F} = 1$. This means $\text{T}$ must be at least $5$ ($\text{T} \ge 5$).

Instead of guessing through hundreds of thousands of combinations by hand, we can state these constraints in Prolog and let its constraint solver deduce the unique solution!

• Constraint Propagation: By applying the carry rules column-by-column (ones, tens, hundreds) instead of testing all permutations at the end, Prolog prunes the search space by over $99.9\%$, finding the answer in a fraction of a millisecond!
• Unique Solution: The unique mathematical solution is: $$734 + 734 = 1468$$ with $\text{T}=7, \text{W}=3, \text{O}=4, \text{F}=1, \text{U}=6, \text{R}=8$.