Puzzle for January 18, 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.5, replace B with D + F (from eq.2): D + F – D = A + E which becomes eq.5a) F = A + E
Hint #2
In eq.4, replace F with A + E (from eq.5a): A + E = A + D Subtract A from each side of the equation above: A + E – A = A + D – A which makes E = D
Hint #3
In eq.3, substitute D for E: C = D + D which makes C = 2×D
Hint #4
Substitute 2×D for C, A + D for F (from eq.4), and D for E in eq.6: A – 2×D + A + D = 2×D + D + D which becomes 2×A – D = 4×D Add D to both sides of the above equation: 2×A – D + D = 4×D + D which makes 2×A = 5×D Divide both sides by 2: 2×A ÷ 2 = 5×D ÷ 2 which makes A = 2½×D
Hint #5
Substitute 2½×D for A, and D for E in eq.5a: F = 2½×D + D which makes F = 3½×D
Hint #6
Substitute 3½×D for F in eq.2: B = D + 3½×D which makes B = 4½×D
Solution
Substitute 2½×D for A, 4½×D for B, 2×D for C, D for E, 3½×D for F in eq.1: 2½×D + 4½×D + 2×D + D + D + 3½×D = 29 which simplifies to 14½×D = 29 Divide both sides of the above equation by 14½: 14½×D ÷ 14½ = 29 ÷ 14½ which means D = 2 making A = 2½×D = 2½ × 2 = 5 B = 4½×D = 4½ × 2 = 9 C = 2×D = 2 × 2 = 4 E = D = 2 F = 3½×D = 3½ × 2 = 7 and ABCDEF = 594227