Puzzle for September 7, 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.3, replace D + E with A + C (from eq.6): C + A + C = A + B which becomes A + 2×C = A + B Subtract A from each side of the above equation: A + 2×C – A = A + B – A which makes 2×C = B
Hint #2
In eq.2, replace B with 2×C: 2×C + C = A which makes 3×C = A
Hint #3
In eq.6, substitute 3×C for A, and C + D for E (from eq.5): 3×C + C = D + C + D which becomes 4×C = C + 2×D Subtract C from each side of the above equation: 4×C – C = C + 2×D – C which becomes 3×C = 2×D Divide both sides by 2: 3×C ÷ 2 = 2×D ÷ 2 which makes 1½×C = D
Hint #4
Substitute 1½×C for D in eq.4: 1½×C = C + F Subtract C from both sides of the equation above: 1½×C – C = C + F – C which makes ½×C = F
Hint #5
Substitute 1½×C for D in eq.5: E = C + 1½×C which makes E = 2½×C
Solution
Substitute 3×C for A, 2×C for B, 1½×C for D, 2½×C for E, and ½×C for F in eq.1: 3×C + 2×C + C + 1½×C + 2½×C + ½×C = 21 which simplifies to 10½×C = 21 Divide both sides of the above equation by 10½: 10½×C ÷ 10½ = 21 ÷ 10½ which means C = 2 making A = 3×C = 3 × 2 = 6 B = 2×C = 2 × 2 = 4 D = 1½×C = 1½ × 2 = 3 E = 2½×C = 2½ × 2 = 5 F = ½×C = ½ × 2 = 1 and ABCDEF = 642351