Puzzle for August 3, 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
eq.5 may be written as: eq.5a) C + E - A + F = D - (B + C)
Hint #2
In eq.5a, replace C + E with D + F (from eq.4), and B + C with A + F (from eq.2): D + F - A + F = D - (A + F) which becomes D + 2×F - A = D - A - F Subtract D from both sides of the above equation: D + 2×F - A - D = D - A - F - D which becomes 2×F - A = -A - F Add A and F to both sides: 2×F - A + A + F = -A - F + A + F which makes 3×F = 0 which means F = 0
Hint #3
Substitute 0 for F in eq.2: A + 0 = B + C which becomes eq.2a) A = B + C
Hint #4
Substitute 0 for F, and B + C for A (from eq.2a) in eq.3: E + 0 = B + C + C which becomes eq.3a) E = B + 2×C
Hint #5
Substitute B + 2×C for E (from eq.3a), (B + C) for A (from eq.2a), and 0 for F in eq.5: C + B + 2×C + 0 - (B + C) = D - B - C which becomes 3×C + B - B - C = D - B - C which becomes 2×C = D - B - C Add B and C to both sides of the equation above: 2×C + B + C = D - B - C + B + C which becomes eq.5b) 3×C + B = D
Hint #6
Substitute 3×C + B for D (from eq.5b), B + 2×C for E (from eq.3a), 0 for F, and B + C for A (from eq.2a) in eq.6: 3×C + B + B + 2×C + 0 - B = B + C + B - C - 0 which simplifies to 5×C + B = 2×B Subtract B from each side of the equation above: 5×C + B - B = 2×B - B which makes 5×C = B
Hint #7
Substitute 5×C for B in eq.5b: 3×C + 5×C = D which makes 8×C = D
Hint #8
Substitute 5×C for B in eq.3a: E = 5×C + 2×C which makes E = 7×C
Hint #9
Substitute 5×C for B in eq.2a: A = 5×C + C which makes A = 6×C
Solution
Substitute 6×C for A, 5×C for B, 8×C for D, 7×C for E, and 0 for F in eq.1: 6×C + 5×C + C + 8×C + 7×C + 0 = 27 which simplifies to 27×C = 27 Divide both sides of the above equation by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 6×C = 6 × 1 = 6 B = 5×C = 5 × 1 = 5 D = 8×C = 8 × 1 = 8 E = 7×C = 7 × 1 = 7 and ABCDEF = 651870