Puzzle for October 30, 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.3, replace D with A + C (from eq.4): C + A + C = A + B which becomes 2×C + A = A + B Subtract A from each side of the equation above: 2×C + A - A = A + B - A which makes 2×C = B
Hint #2
In eq.5, replace B with 2×C: E = 2×C + C which makes E = 3×C
Hint #3
In eq.6, replace B with 2×C: F - C = 2×C - F Add C and F to both sides of the above equation: F - C + C + F = 2×C - F + C + F which makes 2×F = 3×C Divide both sides by 2: 2×F ÷ 2 = 3×C ÷ 2 which makes F = 1½×C
Hint #4
In eq.2, substitute 2×C for B, and 1½×C for F: 2×C = A - 1½×C Add 1½×C to both sides of the above equation: 2×C + 1½×C = A - 1½×C + 1½×C which makes 3½×C = A
Hint #5
Substitute 3½×C for A in eq.4: D = 3½×C + C which makes D = 4½×C
Solution
Substitute 3½×C for A, 2×C for B, 4½×C for D, 3×C for E, and 1½×C for F in eq.1: 3½×C + 2×C + C + 4½×C + 3×C + 1½×C = 31 which simplifies to 15½×C = 31 Divide both sides of the above equation by 15½: 15½×C ÷ 15½ = 31 ÷ 15½ which means C = 2 making A = 3½×C = 3½ × 2 = 7 B = 2×C = 2 × 2 = 4 D = 4½×C = 4½ × 2 = 9 E = 3×C = 3 × 2 = 6 F = 1½×C = 1½ × 2 = 3 and ABCDEF = 742963