Puzzle for November 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
In eq.6, replace A with C + E (from eq.4): C + E - E + F = B + C + E which becomes C + F = B + C + E Subtract C from each side of the equation above: C + F - C = B + C + E - C which becomes eq.6a) F = B + E
Hint #2
In eq.3, replace C with B + D (from eq.2): E = B + D + D which becomes eq.3a) E = B + 2×D
Hint #3
In eq.6a, substitute B + 2×D for E (from eq.3a): F = B + B + 2×D which becomes eq.6b) F = 2×B + 2×D
Hint #4
Substitute B + D for C (from eq.2) and B + 2×D for E (from eq.3a) into eq.4: A = B + D + B + 2×D which becomes eq.4a) A = 2×B + 3×D
Hint #5
Substitute B + D for C (from eq.2), 2×B + 2×D for F (from eq.6b), 2×B + 3×D for A (from eq.4a), and B + 2×D for E (from eq.3a) in eq.5: B + B + D + 2×B + 2×D = 2×B + 3×D + D + B + 2×D which becomes 4×B + 3×D = 3×B + 6×D Subtract 3×D and 3×B from both sides of the above equation: 4×B + 3×D - 3×D - 3×B = 3×B + 6×D - 3×D - 3×B which simplifies to B = 3×D
Hint #6
Substitute (3×D) for B in eq.4a: A = 2×(3×D) + 3×D which becomes A = 6×D + 3×D which makes A = 9×D
Hint #7
Substitute 3×D for B in eq.2: C = 3×D + D which makes C = 4×D
Hint #8
Substitute 3×D for B in eq.3a: E = 3×D + 2×D which makes E = 5×D
Hint #9
Substitute (3×D) for B in eq.6b: F = 2×(3×D) + 2×D which becomes F = 6×D + 2×D which makes F = 8×D
Solution
Substitute 9×D for A, 3×D for B, 4×D for C, 5×D for E, and 8×D for F in eq.1: 9×D + 3×D + 4×D + D + 5×D + 8×D = 30 which simplifies to 30×D = 30 Divide both sides of the above equation by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 9×D = 9 × 1 = 9 B = 3×D = 3 × 1 = 3 C = 4×D = 4 × 1 = 4 E = 5×D = 5 × 1 = 5 F = 8×D = 8 × 1 = 8 and ABCDEF = 934158