Puzzle for November 5, 2019 ( )
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.4, replace B with A + F (from eq.2): A + C = A + F – C In the above equation, subtract A from both sides, and add C to each side: A + C – A + C = A + F – C – A + C which makes 2×C = F
Hint #2
Add C to each side of eq.6: E – C + C = C + F + C which becomes E = 2×C + F In the above equation, replace F with 2×C: E = 2×C + 2×C which makes E = 4×C
Hint #3
Substitute 4×C for E, and 2×C for F in eq.3: C + 4×C + 2×C = A + D which becomes eq.3a) 7×C = A + D Substitute 2×C for F, and 4×C for E in eq.5: D + 2×C = A + 4×C Subtract 2×C from both sides of the above equation: D + 2×C – 2×C = A + 4×C – 2×C which becomes eq.5a) D = A + 2×C
Hint #4
Substitute A + 2×C for D (from eq.5a) in eq.3a: 7×C = A + A + 2×C which becomes 7×C = 2×A + 2×C Subtract 2×C from both sides of the above equation: 7×C – 2×C = 2×A + 2×C – 2×C which makes 5×C = 2×A Divide both sides by 2: 5×C ÷ 2 = 2×A ÷ 2 which makes 2½×C = A
Hint #5
Substitute 2½×C for A in eq.5a: D = 2½×C + 2×C which makes D = 4½×C
Hint #6
Substitute 2½×C for A in eq.4: 2½×C + C = B – C Add C to both sides of the above equation: 2½×C + C + C = B – C + C which means 4½×C = B
Solution
Substitute 2½×C for A, 4½×C for B and D, 4×C for E, and 2×C for F in eq.1: 2½×C + 4½×C + C + 4½×C + 4×C + 2×C = 37 which simplifies to 18½×C = 37 Divide both sides of the equation above by 18½: 18½×C ÷ 18½ = 37 ÷ 18½ which means C = 2 making A = 2½×C = 2½ × 2 = 5 B = D = 4½×C = 4½ × 2 = 9 E = 4×C = 4 × 2 = 8 F = 2×C = 2 × 2 = 4 and ABCDEF = 592984