Puzzle for July 23, 2019 ( )
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.2, replace D + E with B + C (from eq.3): C + B + C = A + B which becomes B + 2×C = A + B Subtract B from both sides of the above equation: B + 2×C - B = A + B - B which makes 2×C = A
Hint #2
In eq.4, replace A with 2×C: D = 2×C + C which makes D = 3×C
Hint #3
In eq.3, substitute A + C for D (from eq.4): B + C = A + C + E Subtract C from each side of the above equation: B + C - C = A + C + E - C which becomes eq.3a) B = A + E
Hint #4
In eq.5, substitute A + E for B (from eq.3a): E - F = A + E - A which becomes E - F = E Subtract E from both sides of the above equation: E - F - E = E - E which makes -F = 0 which means F = 0
Hint #5
Substitute A + E for B (from eq.3a), and 0 for F in eq.6: A + C = A + E + E + 0 which becomes A + C = A + 2×E In the above equation, subtract A from both sides: A + C - A = A + 2×E - A which means C = 2×E Divide both sides by 2: C ÷ 2 = 2×E ÷ 2 which makes ½×C = E
Hint #6
Substitute 2×C for A, and ½×C for E in eq.3a: B = 2×C + ½×C which makes B = 2½×C
Solution
Substitute 2×C for A, 2½×C for B, 3×C for D, ½×C for E, and 0 for F in eq.1: 2×C + 2½×C + C + 3×C + ½×C + 0 = 18 which simplifies to 9×C = 18 Divide both sides of the equation above by 9: 9×C ÷ 9 = 18 ÷ 9 which means C = 2 making A = 2×C = 2 × 2 = 4 B = 2½×C = 2½ × 2 = 5 D = 3×C = 3 × 2 = 6 E = ½×C = ½ × 2 = 1 and ABCDEF = 452610