--- title: "CSP and SWIM+WEAR=RELAX" subject: "Artificial Intelligence" chapter: 2 key: "semester_6/artificial_intelligence:csp-swim-wear-relax" source: https://www.bctengineer.com/q/semester_6/artificial_intelligence/csp-swim-wear-relax site: BCT Engineering --- # CSP and SWIM+WEAR=RELAX Artificial Intelligence · Chapter 2: Problem Solving - Asked in: 2074 Bhadra, 1+6 marks - Video: [L46: CryptArithmetic Problem in Artificial Intelligence | TO+GO = OUT & SEND+MORE= MONEY Solutions](https://www.youtube.com/watch?v=aZblDSAx4cg) — Easy Engineering Classes - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/artificial_intelligence/csp-swim-wear-relax - Whole chapter: https://www.bctengineer.com/md/semester_6/artificial_intelligence/chapter-2 ## Question Define constraint satisfaction problem (CSP). Solve the following crypto-arithmetic problem, where different letters denote different integers and identical letters denote the same integer: SWIM + WEAR = RELAX. ## Short answer CSP is variables with domains and constraints. Solving SWIM+WEAR=RELAX yields S=9, W=4, I=0, M=5, E=3, A=2, R=1, L=7, X=6; 9405+4321=13726. Note the puzzle is not unique; 16 assignments exist. ## Answer #### Constraint satisfaction problem (CSP) A CSP is a problem defined by a set of **variables**, a **domain** of possible values for each, and a set of **constraints** restricting the allowed combinations. Solving it means assigning values to all variables so that every constraint holds (e.g. map colouring, Sudoku, cryptarithmetic). #### SWIM + WEAR = RELAX Variables: S, W, I, M, E, A, R, L, X (9 letters); domain 0-9; all different; S, W, R non-zero. ```text S W I M + W E A R --------- R E L A X ``` Column equations (c1...c4 carries): 1. M + R = X + 10·c1 2. I + A + c1 = A + 10·c2 ⇒ I + c1 = 10·c2 3. W + E + c2 = L + 10·c3 4. S + W + c3 = E + 10·c4 5. R = c4 #### Solving - **R = 1:** it is the carry into the fifth digit, so c4 = 1 and R = 1. - **Column 2:** I + c1 = 10·c2. Either (c2 = 0, I = 0, c1 = 0) or (c2 = 1, I = 9, c1 = 1). The second option leads to no valid assignment (shown by exhaustive search), so take **I = 0, c1 = 0, c2 = 0**. - **Column 1:** M + 1 = X (c1 = 0), so X = M + 1. - **Column 3:** W + E = L + 10·c3; **Column 4:** S + W + c3 = E + 10. - Try c3 = 0: W + E = L and S + W = E + 10. - Take S = 9, W = 4 ⇒ E = 3 (from 9 + 4 = E + 10) and L = 4 + 3 = 7. - Used digits so far: R = 1, I = 0, S = 9, W = 4, E = 3, L = 7. Remaining digits for A, M, X: {2, 5, 6, 8}. - X = M + 1 with both unused: M = 5, X = 6; then **A = 2**. - All different ✓. #### Verification ```text 9 4 0 5 + 4 3 2 1 --------- 1 3 7 2 6 ``` 9405 + 4321 = 13726 ✓ **Answer: S = 9, W = 4, I = 0, M = 5, E = 3, A = 2, R = 1, L = 7, X = 6.** *Note:* a computer search finds 16 valid assignments, e.g. 7604 + 6321 = 13925; each satisfies all the constraints, so the puzzle is not unique. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is R = 1 in SWIM + WEAR = RELAX? A. R is the smallest letter B. R is the carry into the fifth digit, which can only be 1 C. W = 1 D. S = 1 Answer: B. The leading digit of the 5-digit sum is a carry c4 = 1. (Key point: R = 1) 2. What does column 2 (I + A + c1 = A + 10*c2) simplify to? A. I = A B. I = c2 C. I + c1 = 10*c2 D. A = 10 Answer: C. A cancels, leaving I + c1 = 10*c2. (Key point: Column 2) 3. Which option for column 2 is used in the solution? A. I = 9, c1 = 1, c2 = 1 B. I = 1, c1 = 9 C. I = 5, c1 = 5 D. I = 0, c1 = 0, c2 = 0 Answer: D. The I = 9 option leads to no valid assignment, so I = 0 is used. (Key point: Choosing I) 4. What is the verified sum? A. 9405 + 4321 = 13726 B. 9450 + 4321 = 13771 C. 9405 + 4312 = 13717 D. 9504 + 4321 = 13825 Answer: A. S=9, W=4, I=0, M=5, E=3, A=2, R=1, L=7, X=6. 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