Puzzle for March 29, 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.4, replace B + C with A + F (from eq.2): A + E = A + F - D Subtract A from each side of the above equation: A + E - A = A + F - D - A which becomes eq.4a) E = F - D
Hint #2
In eq.3, replace E with F - D (from eq.4a): D + F - D = C + F which becomes F = C + F Subtract F from both sides of the above equation: F - F = C + F - F which makes 0 = C
Hint #3
In eq.5, substitute 0 for C, and F - D for E (from eq.4a): D + F = A + 0 - D + F - D which becomes D + F = A - 2×D + F In the equation above, subtract F from both sides, and add 2×D to both sides: D + F - F + 2×D = A - 2×D + F - F + 2×D which makes 3×D = A
Hint #4
Substitute 0 for C, and 3×D for A in eq.2: B + 0 = 3×D + F which becomes eq.2a) B = 3×D + F
Hint #5
Substitute 0 for C, 3×D for A, and (3×D + F) for B (from eq.2a) in eq.6: 0 + E - 3×D = 3×D - (3×D + F) + D + F which becomes E - 3×D = 4×D - 3×D - F + F which becomes E - 3×D = D Add 3×D to both sides of the above equation: E - 3×D + 3×D = D + 3×D which makes E = 4×D
Hint #6
Substitute 4×D for E in eq.4a: 4×D = F - D Add D to both sides of the above equation: 4×D + D = F - D + D which makes 5×D = F
Hint #7
Substitute 5×D for F in eq.2a: B = 3×D + 5×D which makes B = 8×D
Solution
Substitute 3×D for A, 8×D for B, 0 for C, 4×D for E, and 5×D for F in eq.1: 3×D + 8×D + 0 + D + 4×D + 5×D = 21 which simplifies to 21×D = 21 Divide both sides of the above equation by 21: 21×D ÷ 21 = 21 ÷ 21 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 8×D = 8 × 1 = 8 E = 4×D = 4 × 1 = 4 F = 5×D = 5 × 1 = 5 and ABCDEF = 380145