Puzzle for May 19, 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
eq.6 may be written as: E = (A + B + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (A + B + D) ÷ 3 which becomes eq.6a) 3×E = A + B + D
Hint #2
In eq.6a, replace A + B with C + D (from eq.3): 3×E = C + D + D which becomes 3×E = C + 2×D Subtract 2×D from each side of the equation above: 3×E - 2×D = C + 2×D - 2×D which becomes eq.6b) 3×E - 2×D = C
Hint #3
In eq.2, replace C with 3×E - 2×D (from eq.6b): B + 3×E - 2×D = D + E In the above equation, subtract 3×E from both sides, and add 2×D to both sides: B + 3×E - 2×D - 3×E + 2×D = D + E - 3×E + 2×D which becomes eq.2a) B = 3×D - 2×E
Hint #4
In eq.3, substitute 3×E - 2×D for C (from eq.6b), and 3×D - 2×E for B (from eq.2a): 3×E - 2×D + D = A + 3×D - 2×E which becomes 3×E - D = A + 3×D - 2×E In the equation above, subtract 3×D from both sides, and add 2×E to both sides: 3×E - D - 3×D + 2×E = A + 3×D - 2×E - 3×D + 2×E which becomes eq.3a) 5×E - 4×D = A
Hint #5
Substitute 3×D - 2×E for B (from eq.2a), and 5×E - 4×D for A (from eq.3a) into eq.4: 3×D - 2×E + F = 5×E - 4×D + D which becomes 3×D - 2×E + F = 5×E - 3×D In the above equation, subtract 3×D from both sides, and add 2×E to both sides: 3×D - 2×E + F - 3×D + 2×E = 5×E - 3×D - 3×D + 2×E which becomes eq.4a) F = 7×E - 6×D
Hint #6
Substitute 7×E - 6×D for F (from eq.4a), 5×E - 4×D for A (from eq.3a), and 3×D - 2×E for B (from eq.2a) in eq.5: D - E + 7×E - 6×D = 5×E - 4×D + 3×D - 2×E - D + E which becomes -5×D + 6×E = 4×E - 2×D In the above equation, add 5×D to both sides, and subtract 4×E from both sides: -5×D + 6×E + 5×D - 4×E = 4×E - 2×D + 5×D - 4×E which becomes 2×E = 3×D Divide both sides by 2: 2×E ÷ 2 = 3×D ÷ 2 which makes E = 1½×D
Hint #7
Substitute (1½×D) for E in eq.3a: 5×(1½×D) - 4×D = A which becomes 7½×D - 4×D = A which makes 3½×D = A
Hint #8
Substitute (1½×D) for E in eq.2a: B = 3×D - 2×(1½×D) which becomes B = 3×D - 3×D which makes B = 0
Hint #9
Substitute (1½×D) for E in eq.6b: 3×(1½×D) - 2×D = C which becomes 4½×D - 2×D = C which makes 2½×D = C
Hint #10
Substitute 1½×D for E in eq.4a: F = 7×(1½×D) - 6×D which becomes F = 10½×D - 6×D which makes F = 4½×D
Solution
Substitute 3½×D for A, 0 for B, 2½×D for C, 1½×D for E, and 4½×D for F in eq.1: 3½×D + 0 + 2½×D + D + 1½×D + 4½×D = 26 which simplifies to 13×D = 26 Divide both sides of the above equation by 13: 13×D ÷ 13 = 26 ÷ 13 which means D = 2 making A = 3½×D = 3½ × 2 = 7 C = 2½×D = 2½ × 2 = 5 E = 1½×D = 1½ × 2 = 3 F = 4½×D = 4½ × 2 = 9 and ABCDEF = 705239