Puzzle for January 26, 2022 ( )
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 C + D with B (from eq.2): F = B + B which makes eq.4a) F = 2×B
Hint #2
In eq.6, replace F with 2×B (from eq.4a), and E with A + B + D (from eq.5): B + 2×B = D + A + B + D which becomes 3×B = A + B + 2×D Subtract B and 2×D from each side of the equation above: 3×B – B – 2×D = A + B + 2×D – B – 2×D which becomes eq.6a) 2×B – 2×D = A
Hint #3
In eq.3, substitute 2×B – 2×D for A (from eq.6a): D = 2×B – 2×D + C Add 2×D to both sides of the above equation: D + 2×D = 2×B – 2×D + C + 2×D which becomes eq.3a) 3×D = 2×B + C
Hint #4
Substitute (C + D) for B (from eq.2) into eq.3a: 3×D = 2×(C + D) + C which is equivalent to 3×D = 2×C + 2×D + C which becomes 3×D = 3×C + 2×D Subtract 2×D from both sides of the equation above: 3×D – 2×D = 3×C + 2×D – 2×D which makes D = 3×C
Hint #5
Substitute 3×C for D in eq.2: B = C + 3×C which makes B = 4×C
Hint #6
Substitute (4×C) for B in eq.4a: F = 2×(4×C) which makes F = 8×C
Hint #7
Substitute 3×C for D in eq.3: 3×C = A + C Subtract C from each side of the equation above: 3×C – C = A + C – C which makes 2×C = A
Hint #8
Substitute 2×C for A, 4×C for B, and 3×C for D in eq.5: E = 2×C + 4×C + 3×C which makes E = 9×C
Solution
Substitute 2×C for A, 4×C for B, 3×C for D, 9×C for E, and 8×C for F in eq.1: 2×C + 4×C + C + 3×C + 9×C + 8×C = 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 = 2×C = 2 × 1 = 2 B = 4×C = 4 × 1 = 4 D = 3×C = 3 × 1 = 3 E = 9×C = 9 × 1 = 9 F = 8×C = 8 × 1 = 8 and ABCDEF = 241398