Puzzle for September 14, 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.3, replace C with E + F (from eq.2): A = E + F + E which becomes eq.3a) A = 2×E + F
Hint #2
In eq.5, replace A with 2×E + F (from eq.3a), and D with B + F (from eq.4): B + E = 2×E + F - B + B + F which becomes B + E = 2×E + 2×F Subtract E from each side of the equation above: B + E - E = 2×E + 2×F - E which becomes eq.5a) B = E + 2×F
Hint #3
In eq.4, substitute E + 2×F for B (from eq.5a): D = E + 2×F + F which becomes eq.4a) D = E + 3×F
Hint #4
In eq.6, substitute E + 2×F for B (from eq.5a), E + 3×F for D (from eq.4a), 2×E + F for A (from eq.3a), and E + F for C (from eq.2): E + 2×F + E + 3×F = 2×E + F + E + F + E which becomes 2×E + 5×F = 4×E + 2×F Subtract 2×E and 2×F from each side of the equation above: 2×E + 5×F - 2×E - 2×F = 4×E + 2×F - 2×E - 2×F which makes 3×F = 2×E Divide both sides by 2: 3×F ÷ 2 = 2×E ÷ 2 which makes 1½×F = E
Hint #5
Substitute (1½×F) for E in eq.3a: A = 2×(1½×F) + F which becomes A = 3×F + F which makes A = 4×F
Hint #6
Substitute 1½×F for E in eq.5a: B = 1½×F + 2×F which makes B = 3½×F
Hint #7
Substitute 1½×F for E in eq.2: C = 1½×F + F which makes C = 2½×F
Hint #8
Substitute 1½×F for E in eq.4a: D = 1½×F + 3×F which makes D = 4½×F
Solution
Substitute 4×F for A, 3½×F for B, 2½×F for C, 4½×F for D, and 1½×F for E in eq.1: 4×F + 3½×F + 2½×F + 4½×F + 1½×F + F = 34 which simplifies to 17×F = 34 Divide both sides of the above equation by 17: 17×F ÷ 17 = 34 ÷ 17 which means F = 2 making A = 4×F = 4 × 2 = 8 B = 3½×F = 3½ × 2 = 7 C = 2½×F = 2½ × 2 = 5 D = 4½×F = 4½ × 2 = 9 E = 1½×F = 1½ × 2 = 3 and ABCDEF = 875932