Puzzle for October 5, 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.5, replace A with D + F (from eq.3): 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
Subtract D from each side of eq.2: B - D = C + D - D which becomes eq.2a) B - D = C
Hint #3
In eq.4, replace C with B - D (from eq.2a), and E with 2×D: F = B - D + 2×D which becomes eq.4a) F = B + D
Hint #4
In eq.3, substitute B + D for F (from eq.4a): A = D + B + D which becomes eq.3a) A = 2×D + B
Hint #5
eq.6 may be written as: C = (A + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (A + D + F) ÷ 3 which becomes eq.6a) 3×C = A + D + F
Hint #6
Substitute (B - D) for C (from eq.2a), 2×D + B for A (from eq.3a), and B + D for F (from eq.4a) in eq.6a: 3×(B - D) = 2×D + B + D + B + D which becomes 3×B - 3×D = 4×D + 2×B In the above equation, add 3×D to both sides, and subtract 2×B from both sides: 3×B - 3×D + 3×D - 2×B = 4×D + 2×B + 3×D - 2×B which simplifies to B = 7×D
Hint #7
Substitute 7×D for B in eq.2a: 7×D - D = C which makes 6×D = C
Hint #8
Substitute 7×D for B in eq.3a: A = 2×D + 7×D which makes A = 9×D
Hint #9
Substitute 7×D for B in eq.4a: F = 7×D + D which makes F = 8×D
Solution
Substitute 9×D for A, 7×D for B, 6×D for C, 2×D for E, and 8×D for F in eq.1: 9×D + 7×D + 6×D + D + 2×D + 8×D = 33 which simplifies to 33×D = 33 Divide both sides of the above equation by 33: 33×D ÷ 33 = 33 ÷ 33 which means D = 1 making A = 9×D = 9×1 = 9 B = 7×D = 7×1 = 7 C = 6×D = 6×1 = 6 E = 2×D = 2×1 = 2 F = 8×D = 8×1 = 8 and ABCDEF = 976128