Puzzle for September 8, 2022 ( )
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.6, replace D with A + B (from eq.2): B + A + B = A + C which becomes 2×B + A = A + C Subtract A from each side of the equation above: 2×B + A - A = A + C - A which makes eq.6a) 2×B = C
Hint #2
In eq.5, replace C with 2×B: E + F = B + 2×B which becomes eq.5a) E + F = 3×B
Hint #3
Add F to both sides of eq.3: E + F = A + F + F which becomes eq.3a) E + F = A + 2×F
Hint #4
In eq.3a, substitute 3×B for E + F (from eq.5a), and B + F for A (from eq.4): 3×B = B + F + 2×F which becomes 3×B = B + 3×F Subtract B from each side of the above equation: 3×B - B = B + 3×F - B which makes 2×B = 3×F Divide both sides by 2: 2×B ÷ 2 = 3×F ÷ 2 which makes B = 1½×F
Hint #5
Substitute (1½×F) for B in eq.6a: 2×(1½×F) = C which makes 3×F = C
Hint #6
Substitute 1½×F for B, and 3×F for C in eq.5: E + F = 1½×F + 3×F which becomes E + F = 4½×F Subtract F from both sides of the above equation: E + F - F = 4½×F - F which makes E = 3½×F
Hint #7
Substitute 3½×F for E in eq.3: 3½×F = A + F Subtract F from both sides of the equation above: 3½×F - F = A + F - F which makes 2½×F = A
Hint #8
Substitute 2½×F for A, and 1½×F for B in eq.2: D = 2½×F + 1½×F which makes D = 4×F
Solution
Substitute 2½×F for A, 1½×F for B, 3×F for C, 4×F for D, and 3½×F for E in eq.1: 2½×F + 1½×F + 3×F + 4×F + 3½×F + F = 31 which simplifies to 15½×F = 31 Divide both sides of the above equation by 15½: 15½×F ÷ 15½ = 31 ÷ 15½ which means F = 2 making A = 2½×F = 2½ × 2 = 5 B = 1½×F = 1½ × 2 = 3 C = 3×F = 3 × 2 = 6 D = 4×F = 4 × 2 = 8 E = 3½×F = 3½ × 2 = 7 and ABCDEF = 536872