Puzzle for April 4, 2023 ( )
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 E with A + C (from eq.2): C + A + C = A - C + F which becomes 2×C + A = A - C + F In the above equation, subtract A from both sides, and add C to both sides: 2×C + A - A + C = A - C + F - A + C which simplifies to 3×C = F
Hint #2
In eq.3, replace F with 3×C: D = C + 3×C which makes D = 4×C
Hint #3
In eq.4, replace F with 3×C, and D with 4×C: 3×C - A = A - 4×C Add A and 4×C to both sides of the above equation: 3×C - A + A + 4×C = A - 4×C + A + 4×C which makes 7×C = 2×A Divide both sides by 2: 7×C ÷ 2 = 2×A ÷ 2 which makes 3½×C = A
Hint #4
In eq.2, substitute 3½×C for A: E = 3½×C + C which makes E = 4½×C
Hint #5
Substitute 4×C for D, 4½×C for E, and 3½×C for A in eq.6: 4×C - 4½×C = 3½×C + B - 4×C which becomes -½×C = -½×C + B Add ½×C to both sides of the equation above: -½×C + ½×C = -½×C + B + ½×C which makes 0 = B
Solution
Substitute 3½×C for A, 0 for B, 4×C for D, 4½×C for E, and 3×C for F in eq.1: 3½×C + 0 + C + 4×C + 4½×C + 3×C = 32 which simplifies to 16×C = 32 Divide both sides of the above equation by 16: 16×C ÷ 16 = 32 ÷ 16 which means C = 2 making A = 3½×C = 3½ × 2 = 7 D = 4×C = 4 × 2 = 8 E = 4½×C = 4½ × 2 = 9 F = 3×C = 3 × 2 = 6 and ABCDEF = 702896