Puzzle for December 15, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Subtract the left and right sides of eq.5 from the left and right sides of eq.4, respectively: D + E - (B + E) = A + F - (C + F) which becomes D + E - B - E = A + F - C - F which becomes D - B = A - C Add B and C to both sides of the above equation: D - B + B + C = A - C + B + C which becomes eq.4a) D + C = A + B
Hint #2
In eq.2, replace A + B with D + C (from eq.4a): eq.2a) F = D + C
Hint #3
In eq.6, substitute (D + C) for F (from eq.2a): C - (D + C) = D - C which becomes C - D - C = D - C which becomes -D = D - C Add D and C to both sides of the equation above: -D + D + C = D - C + D + C which makes C = 2×D
Hint #4
Substitute 2×D for C in eq.2a: F = D + 2×D which makes F = 3×D
Hint #5
Substitute 2×D for C, and 3×D for F in eq.5: B + E = 2×D + 3×D which becomes eq.5a) B + E = 5×D
Hint #6
Substitute 3×D for F in eq.3: E = B + D + 3×D which becomes eq.3a) E = B + 4×D
Hint #7
Substitute B + 4×D for E (from eq.3a) in eq.5a: B + B + 4×D = 5×D which becomes 2×B + 4×D = 5×D Subtract 4×D from each side of the equation above: 2×B + 4×D - 4×D = 5×D - 4×D which makes 2×B = D Divide both sides by 2: 2×B ÷ 2 = D ÷ 2 which makes B = ½×D
Hint #8
Substitute ½×D for B in eq.3a: E = ½×D + 4×D which makes E = 4½×D
Hint #9
Substitute 3×D for F, and ½×D for B in eq.2: 3×D = A + ½×D Subtract ½×D from each side of the above equation: 3×D - ½×D = A + ½×D - ½×D which makes 2½×D = A
Solution
Substitute 2½×D for A, ½×D for B, 2×D for C, 4½×D for E, and 3×D for F in eq.1: 2½×D + ½×D + 2×D + D + 4×½×D + 3×D = 27 which simplifies to 13½×D = 27 Divide both sides of the above equation by 13½: 13½×D ÷ 13½ = 27 ÷ 13½ which means D = 2 making A = 2½×D = 2½ × 2 = 5 B = ½×D = ½ × 2 = 1 C = 2×D = 2 × 2 = 4 E = 4½×D = 4½ × 2 = 9 F = 3×D = 3 × 2 = 6 and ABCDEF = 514296