Puzzle for March 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.3, replace C with D + F (from eq.2): E + F = D + F + D which becomes E + F = 2×D + F Subtract F from each side of the equation above: E + F - F = 2×D + F - F which makes E = 2×D
Hint #2
eq.5 may be re-written as: F - D = A + B + E + D In the above equation, replace A + B + E with F - B (from eq.4): F - D = F - B + D Subtract F from both sides, and add D and B to both sides: F - D - F + D + B = F - B + D - F + D + B which makes B = 2×D
Hint #3
In eq.5, substitute 2×D for B and E: F - D = A + 2×D + D + 2×D which becomes F - D = A + 5×D Add D to both sides of the above equation: F - D + D = A + 5×D + D which becomes eq.5a) F = A + 6×D
Hint #4
In eq.6, substitute 2×D for B: A + 2×D + D = F - A - D which becomes A + 3×D = F - A - D Add A and D to both sides of the equation above: A + 3×D + A + D = F - A - D + A + D which becomes eq.6a) 2×A + 4×D = F
Hint #5
Substitute A + 6×D for F (from eq.5a) into eq.6a: 2×A + 4×D = A + 6×D Subtract A and 4×D from each side of the above equation: 2×A + 4×D - A - 4×D = A + 6×D - A - 4×D which makes A = 2×D
Hint #6
Substitute 2×D for A in eq.5a: F = 2×D + 6×D which makes F = 8×D
Hint #7
Substitute 8×D for F in eq.2: C = D + 8×D which makes C = 9×D
Solution
Substitute 2×D for A and B and E, 9×D for C, and 8×D for F in eq.1: 2×D + 2×D + 9×D + D + 2×D + 8×D = 24 which simplifies to 24×D = 24 Divide both sides of the above equation by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = B = E = 2×D = 2 × 1 = 2 C = 9×D = 9 × 1 = 9 F = 8×D = 8 × 1 = 8 and ABCDEF = 229128