Puzzle for April 13, 2023 ( )
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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.
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Hint #1
In eq.5, replace C with D + E (from eq.2): E + F = B + D + E - E which becomes eq.5a) E + F = B + D
Hint #2
Add F and A to both sides of eq.3: B - F + F + A = E - A + F + A which becomes eq.3a) B + A = E + F
Hint #3
In eq.5a, replace E + F with B + A (from eq.3a): B + A = B + D Subtract B from each side of the equation above: B + A - B = B + D - B which makes A = D
Hint #4
In eq.4, replace D with A: F - A = A + A which becomes F - A = 2×A Add A to both sides of the above equation: F - A + A = 2×A + A which makes F = 3×A
Hint #5
In eq.3a, substitute 3×A for F: B + A = E + 3×A Subtract A from each side of the above equation: B + A - A = E + 3×A - A which becomes eq.3b) B = E + 2×A
Hint #6
Substitute 3×A for F, and E + 2×A for B (from eq.3b) in eq.5: E + 3×A = E + 2×A + C - E which becomes E + 3×A = 2×A + C Subtract 2×A from both sides of the equation above: E + 3×A - 2×A = 2×A + C - 2×A which becomes eq.5b) E + A = C
Hint #7
Substitute 3×A for F, E + A for C (from eq.5b), and A for D in eq.6: A + 3×A = E + A + A + E - 3×A which becomes 4×A = 2×E - A Add A to both sides of the above equation: 4×A + A = 2×E - A + A which makes 5×A = 2×E Divide both sides by 2: 5×A ÷ 2 = 2×E ÷ 2 which makes 2½×A = E
Hint #8
Substitute 2½×A for E in eq.5b: 2½×A + A = C which makes 3½×A = C
Hint #9
Substitute 2½×A for E in eq.3b: B = 2½×A + 2×A which makes B = 4½×A
Solution
Substitute 4½×A for B, 3½×A for C, A for D, 2½×A for E, and 3×A for F in eq.1: A + 4½×A + 3½×A + A + 2½×A + 3×A = 31 which simplifies to 15½×A = 31 Divide both sides of the above equation by 15½: 15½×A ÷ 15½ = 31 ÷ 15½ which means A = 2 making B = 4½×A = 4½ × 2 = 9 C = 3½×A = 3½ × 2 = 7 D = A = 2 E = 2½×A = 2½ × 2 = 5 F = 3×A = 3 × 2 = 6 and ABCDEF = 297256