Puzzle for February 18, 2020 ( )
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 with B + C (from eq.4): C + B + C = A + B which becomes B + 2×C = A + B Subtract B from each side of the equation above: B + 2×C – B = A + B – B which makes eq.2a) 2×C = A
Hint #2
In eq.3, replace F with D + E (from eq.5): E + D + E = C + D which becomes D + 2×E = C + D Subtract D from each side of the above equation: D + 2×E – D = C + D – D which makes 2×E = C
Hint #3
In eq.2a, substitute (2×E) for C: 2×(2×E) = A which makes 4×E = A
Hint #4
Substitute 4×E for A, and B + C for D (from eq.4) in eq.6: B + B + C = 4×E + E which becomes eq.6a) 2×B + C = 5×E
Hint #5
Substitute 2×E for C in eq.6a: 2×B + 2×E = 5×E Subtract 2×E from each side of the equation above: 2×B + 2×E – 2×E = 5×E – 2×E which makes 2×B = 3×E Divide both sides by 2: 2×B ÷ 2 = 3×E ÷ 2 which makes B = 1½×E
Hint #6
Substitute 1½×E for B, and 2×E for C in eq.4: D = 1½×E + 2×E which makes D = 3½×E
Hint #7
Substitute 2×E for C, and 3½×E for D in eq.3: E + F = 2×E + 3½×E which becomes E + F = 5½×E Subtract E from both sides of the equation above: E + F – E = 5½×E – E F = 4½×E
Solution
Substitute 4×E for A, 1½×E for B, 2×E for C, 3½×E for D, and 4½×E for F in eq.1: 4×E + 1½×E + 2×E + 3½×E + E + 4½×E = 33 which simplifies to 16½×E = 33 Divide both sides of the equation above by 16½: 16½×E ÷ 16½ = 33 ÷ 16½ which means E = 2 making A = 4×E = 4 × 2 = 8 B = 1½×E = 1½ × 2 = 3 C = 2×E = 2 × 2 = 4 D = 3½×E = 3½ × 2 = 7 F = 4½×E = 4½ × 2 = 9 and ABCDEF = 834729