Puzzle for April 20, 2020 ( )
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 F with A + D (from eq.6): D + A + D = A + C which becomes A + 2×D = A + C Subtract A from each side of the equation above: A + 2×D – A = A + C – A which makes 2×D = C
Hint #2
In eq.2, replace C with 2×D: B = 2×D + D which makes B = 3×D
Hint #3
In eq.5, substitute 2×D for C, and 3×D for B: E – 2×D = 3×D + D which becomes E – 2×D = 4×D Add 2×D to both sides of the above equation: E – 2×D + 2×D = 4×D + 2×D which makes E = 6×D
Hint #4
Substitute 2×D for C, and 6×D for E in eq.3: 2×D + 6×D = A which makes 8×D = A
Hint #5
Substitute 8×D for A in eq.6: F = 8×D + D which makes F = 9×D
Solution
Substitute 8×D for A, 3×D for B, 2×D for C, 6×D for E, and 9×D for F in eq.1: 8×D + 3×D + 2×D + D + 6×D + 9×D = 29 which simplifies to 29×D = 29 Divide both sides of the above equation by 29: 29×B ÷ 29 = 29 ÷ 29 which means D = 1 making A = 8×D = 8 × 1 = 8 B = 3×D = 3 × 1 = 3 C = 2×D = 2 × 1 = 2 E = 6×D = 6 × 1 = 6 F = 9×D = 9 × 1 = 9 and ABCDEF = 832169