Cryptarithmetic RIGHT+RIGHT=WRONG

This is a constraint satisfaction problem.

• Variables: R, I, G, H, T, W, O, N
• Domain: {0, 1, ..., 9}
• Constraints: all letters different; R and W (leading letters) non-zero; the addition must be arithmetically correct.

      R I G H T
    + R I G H T
    -----------
      W R O N G

The result has the same number of digits, so there is no carry out of the leftmost column.

Column equations (c1...c4 = carries, each 0 or 1)

1. T + T = G + 10·c1
2. H + H + c1 = N + 10·c2
3. G + G + c2 = O + 10·c3
4. I + I + c3 = R + 10·c4
5. R + R + c4 = W (W ≤ 9)

Solving by constraint propagation and backtracking

• Column 1: G = 2T mod 10, so G is even and c1 = 1 if T ≥ 5.
• Column 5: W = 2R + c4 ≤ 9 ⇒ R ≤ 4, and W is even if c4 = 0.
• Choose T = 5: 5 + 5 = 10 ⇒ G = 0, c1 = 1.
• Column 2: H + H + 1 = N + 10·c2. Choose H = 6: 13 ⇒ N = 3, c2 = 1.
• Column 3: G + G + c2 = 0 + 0 + 1 = 1 ⇒ O = 1, c3 = 0.
• Column 4: I + I + 0 = R + 10·c4. Choose I = 2: 4 ⇒ R = 4, c4 = 0.
• Column 5: R + R + c4 = 8 ⇒ W = 8.
• Check distinctness: R = 4, I = 2, G = 0, H = 6, T = 5, W = 8, O = 1, N = 3, all different and R, W ≠ 0 ✓.

Verification

      4 2 0 6 5
    + 4 2 0 6 5
    -----------
      8 4 1 3 0

42065 × 2 = 84130 = WRONG ✓

Answer: R = 4, I = 2, G = 0, H = 6, T = 5, W = 8, O = 1, N = 3 (42065 + 42065 = 84130).

Note: the puzzle has several valid assignments (a computer search gives 11 in all, e.g. RIGHT = 47065, 10678, 15648, 31854). Any one that satisfies all constraints is correct.