Puzzle for September 1, 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.2, replace C with A + B (from eq.3): B = A + B - D + E In the above equation, subtract B from both sides, and add D to both sides: B - B + D = A + B - D + E - B + D which becomes eq.2a) D = A + E
Hint #2
In eq.5, substitute (A + E) for D (from eq.2a): E = A + C - (A + E) + F which becomes E = A + C - A - E + F which becomes E = C - E + F Add E to both sides of the above equation: E + E = C - E + F + E which becomes eq.5a) 2×E = C + F
Hint #3
In eq.4, substitute A + E for D (from eq.2a), and 2×E for C + F (from eq.5a): A + E = 2×E - A In the above equation, subtract E from both sides, and add A to both sides: A + E - E + A = 2×E - A - E + A which simplifies to eq.4a) 2×A = E
Hint #4
Substitute 2×A for E in eq.2a: D = A + 2×A which makes eq.2b) D = 3×A
Hint #5
Substitute (2×A) for E, and B + C for F (from eq.6) in eq.5a: 2×(2×A) = C + B + C which becomes eq.5b) 4×A = 2×C + B
Hint #6
Substitute (A + B) for C (from eq.3) in eq.5b: 4×A = 2×(A + B) + B which becomes 4×A = 2×A + 2×B + B which becomes 4×A = 2×A + 3×B Subtract 2×A from both sides of the above equation: 4×A - 2×A = 2×A + 3×B - 2×A which makes 2×A = 3×B Divide both sides by 2: 2×A ÷ 2 = 3×B ÷ 2 which makes A = 1½×B
Hint #7
Substitute (1½×B) for A in eq.4a: 2×(1½×B) = E which makes 3×B = E
Hint #8
Substitute (1½×B) for A in eq.2b: D = 3×(1½×B) which makes D = 4½×B
Hint #9
Substitute (1½×B) for A in eq.5b: 4×(1½×B) = 2×C + B which becomes 6×B = 2×C + B Subtract B from both sides of the equation above: 6×B - B = 2×C + B - B which makes 5×B = 2×C Divide both sides by 2: 5×B ÷ 2 = 2×C ÷ 2 which makes 2½×B = C
Hint #10
Substitute 2½×B for C in eq.6: F = B + 2½×B which makes F = 3½×B
Solution
Substitute 1½×B for A, 2½×B for C, 4½×B for D, 3×B for E, and 3½×B for F in eq.1: 1½×B + B + 2½×B + 4½×B + 3×B + 3½×B = 32 which simplifies to 16×B = 32 Divide both sides of the above equation by 16: 16×B ÷ 16 = 32 ÷ 16 which means B = 2 making A = 1½×B = 1½ × 2 = 3 C = 2½×B = 2½ × 2 = 5 D = 4½×B = 4½ × 2 = 9 E = 3×B = 3 × 2 = 6 F = 3½×B = 3½ × 2 = 7 and ABCDEF = 325967