Puzzle for May 24, 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, substitute (A + C) for D (from eq.2): A - C = C - (A + C) which becomes A - C = C - A - C which becomes A - C = -A Add C and A to both sides of the above equation: A - C + C + A = -A + C + A which makes 2×A = C
Hint #2
In eq.2, replace C with 2×A: D = A + 2×A which makes D = 3×A
Hint #3
In eq.6, replace C with 2×A, and D with 3×A: 2×A + 3×A = B - 2×A + F which becomes 5×A = B - 2×A + F Add 2×A to both sides of the above equation: 5×A + 2×A = B - 2×A + F + 2×A which becomes eq.6a) 7×A = B + F
Hint #4
In eq.3, substitute 3×A for D: B + 3×A = A + F Subtract A from each side of the equation above: B + 3×A - A = A + F - A which becomes eq.3a) B + 2×A = F
Hint #5
Substitute B + 2×A for F (from eq.3a) into eq.6a: 7×A = B + B + 2×A which becomes 7×A = 2×B + 2×A Subtract 2×A from each side of the above equation: 7×A - 2×A = 2×B + 2×A - 2×A which makes 5×A = 2×B Divide both sides by 2: 5×A ÷ 2 = 2×B ÷ 2 which makes 2½×A = B
Hint #6
Substitute 2½×A for B in eq.3a: 2½×A + 2×A = F which makes 4½×A = F
Hint #7
Substitute 2½×A for B in eq.4: E - A = A + 2½×A which becomes E - A = 3½×A Add A to both sides of the above equation: E - A + A = 3½×A + A which makes E = 4½×A
Solution
Substitute 2½×A for B, 2×A for C, 3×A for D, and 4½×A for E and F in eq.1: A + 2½×A + 2×A + 3×A + 4½×A + 4½×A = 35 which simplifies to 17½×A = 35 Divide both sides of the above equation by 17½: 17½×A ÷ 17½ = 35 ÷ 17½ which means A = 2 making B = 2½×A = 2½ × 2 = 5 C = 2×A = 2 × 2 = 4 D = 3×A = 3 × 2 = 6 E = F = 4½×A = 4½ × 2 = 9 and ABCDEF = 254699