Puzzle for November 22, 2022 ( )
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
In eq.3, replace F with C + E (from eq.2): D + E = C + E - D In the above equation, subtract E from both sides, and add D to both sides: D + E - E + D = C + E - D - E + D which makes 2×D = C
Hint #2
In eq.4, replace F with C + E (from eq.2): B + E = C + C + E which becomes B + E = 2×C + E Subtract E from each side of the equation above: B + E - E = 2×C + E - E which makes eq.4a) B = 2×C
Hint #3
In eq.4a, substitute (2×D) for C: B = 2×(2×D) which makes B = 4×D
Hint #4
eq.6 may be written as: E = (C + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (C + D + F) ÷ 3 which becomes eq.6a) 3×E = C + D + F Add D to both sides of eq.3: D + E + D = F - D + D which becomes eq.3a) 2×D + E = F
Hint #5
Substitute 2×D for C, and 2×D + E for F (from eq.3a) in eq.6a: 3×E = 2×D + D + 2×D + E which becomes 3×E = 5×D + E Subtract E from each side of the above equation: 3×E - E = 5×D + E - E which makes 2×E = 5×D Divide both sides by 2: 2×E ÷ 2 = 5×D ÷ 2 which makes E = 2½×D
Hint #6
Substitute 2½×D for E in eq.3a: 2×D + 2½×D = F which makes 4½×D = F
Hint #7
Substitute 2×D for C, 2½×D for E, and 4½×D for F in eq.5: 2×D + D = 2½×D + 4½×D - A which becomes 3×D = 7×D - A In the above equation, subtract 3×D from both sides, and add A to both sides: 3×D - 3×D + A = 7×D - A - 3×D + A which makes A = 4×D
Solution
Substitute 4×D for A and B, 2×D for C, 2½×D for E, and 4½×D for F in eq.1: 4×D + 4×D + 2×D + D + 2½×D + 4½×D = 36 which simplifies to 18×D = 36 Divide both sides of the above equation by 18: 18×D ÷ 18 = 36 ÷ 18 which means D = 2 making A = B = 4×D = 4 × 2 = 8 C = 2×D = 2 × 2 = 4 E = 2½×D = 2½ × 2 = 5 F = 4½×D = 4½ × 2 = 9 and ABCDEF = 884259