Puzzle for November 7, 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.3, replace F with B + E (from eq.4): A + E = B + B + E which becomes A + E = 2×B + E Subtract E from both sides of the above equation: A + E – E = 2×B + E – E which makes eq.3a) A = 2×B
Hint #2
In eq.6, replace A with 2×B, and replace B + C with D (from eq.2): D + E + F = 2×B + D Subtract D from both sides of the above equation: D + E + F – D = 2×B + D – D which becomes eq.6a) E + F = 2×B
Hint #3
In eq.6a, replace F with B + E (from eq.4): E + B + E = 2×B which becomes B + 2×E = 2×B Subtract B from both sides of the above equation: B + 2×E – B = 2×B – B which makes 2×E = B
Hint #4
In eq.3a, substitute (2×E) for B: A = 2×(2×E) which makes A = 4×E
Hint #5
In eq.6a, substitute (2×E) for B: E + F = 2×(2×E) which is equivalent to E + F = 4×E Subtract E from both sides of the above equation: F = 4×E – E which makes F = 3×E
Hint #6
Substitute 2×E for B in eq.2: eq.2a) 2×E + C = D Substitute 4×E for A, 2×E for B, and 3×E for F in eq.5: C + D = 4×E – 2×E + 3×E which becomes eq.5a) C + D = 5×E
Hint #7
Substitute 2×E + C for D (from eq.2a) in eq.5a: C + 2×E + C = 5×E which becomes 2×C + 2×E = 5×E Subtract 2×E from each side of the above equation: 2×C + 2×E – 2×E = 5×E – 2×E which means 2×C = 3×E Divide both sides by 2: 2×C ÷ 2 = 3×E ÷ 2 which makes C = 1½×E
Hint #8
Substitute 1½×E for C in eq.2a: 2×E + 1½×E = D which makes 3½×E = D
Solution
Substitute 4×E for A, 2×E for B, 1½×E for C, 3½×E for D, and 3×E for F in eq.1: 4×E + 2×E + 1½×E + 3½×E + E + 3×E = 30 which simplifies to 15×E = 30 Divide both sides of the above equation by 15: 15×E ÷ 15 = 30 ÷ 15 which means E = 2 making A = 4×E = 4 × 2 = 8 B = 2×E = 2 × 2 = 4 C = 1½×E = 1½ × 2 = 3 D = 3½×E = 3½ × 2 = 7 F = 3×E = 3 × 2 = 6 and ABCDEF = 843726