Cryptarithmetic ONE+ONE+TWO=FOUR

This is a CSP with variables O, N, E, T, W, F, U, R, domain {0..9}, all values different, and leading letters O, T, F non-zero.

      O N E
      O N E
    + T W O
    -------
    F O U R

Column equations (c1, c2, c3 = carries)

1. E + E + O = R + 10·c1
2. N + N + W + c1 = U + 10·c2
3. O + O + T + c2 = O + 10·c3 ⇒ O + T + c2 = 10·c3
4. F = c3

Reasoning

• F = 1: F is only a carry. O + T + c2 ≤ 9 + 8 + 2 = 19, so c3 ≤ 1, and F ≠ 0 means F = 1.
• Then O + T + c2 = 10, so T = 10 − O − c2.
• Choose O = 6. Column 1 becomes 2E + 6 = R + 10·c1.
  • Try E = 2: 4 + 6 = 10 ⇒ R = 0, c1 = 1.
• Column 2 can give a carry c2 of 0, 1 or 2; try c2 = 1. Then T = 10 − 6 − 1 = 3.
• Column 2: 2N + W + 1 = U + 10 (c2 = 1). Choose N = 4: 8 + W + 1 = U + 10 ⇒ U = W − 1. Take W = 8, U = 7 (digits 7 and 8 are still unused).
• Check that all digits differ: O = 6, N = 4, E = 2, T = 3, W = 8, F = 1, U = 7, R = 0. Yes, all eight are different.

Verification

      6 4 2
      6 4 2
    + 3 8 6
    -------
    1 6 7 0

642 + 642 + 386 = 1670 ✓

Answer: O = 6, N = 4, E = 2, T = 3, W = 8, F = 1, U = 7, R = 0 (642 + 642 + 386 = 1670).

Note: F = 1 and O + T + c2 = 10 hold in every solution, but the other letters are not forced uniquely. A computer search finds 107 valid assignments (distinct digits, no leading zero). Other correct answers are 392 + 392 + 563 = 1347, 428 + 428 + 574 = 1430 and 658 + 658 + 376 = 1692. Any assignment that satisfies all constraints is correct.