Puzzle for February 11, 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
Add C to both sides of eq.2: D + E + C = F - C + C which becomes eq.2a) D + E + C = F
Hint #2
In eq.5, replace F with D + E + C (from eq.2a): C + D + E + C = B - C + E which becomes 2×C + D + E = B - C + E In the above equation, subtract E from both sides, and add C to both sides: 2×C + D + E - E + C = B - C + E - E + C which becomes eq.5a) 3×C + D = B
Hint #3
In eq.4, replace D + E with F - C (from eq.2): B + F = A + F - C In the above equation, subtract F from both sides, and add C to both sides: B + F - F + C = A + F - C - F + C which becomes eq.4a) B + C = A
Hint #4
In eq.4a, substitute 3×C + D for B (from eq.5a): 3×C + D + C = A which becomes eq.4b) 4×C + D = A
Hint #5
Substitute 4×C + D for A (from eq.4b), and 3×C + D for B (from eq.5a) in eq.3: C + E = 4×C + D + 3×C + D + D which becomes C + E = 7×C + 3×D Subtract C from each side of the equation above: C + E - C = 7×C + 3×D - C which becomes eq.3a) E = 6×C + 3×D
Hint #6
Substitute 6×C + 3×D for E (from eq.3a) in eq.2a: D + 6×C + 3×D + C = F which becomes eq.2b) 4×D + 7×C = F
Hint #7
In eq.6, substitute 4×C + D for A (from eq.4b), 3×C + D for B (from eq.5a), (6×C + 3×D) for E (from eq.3a), and 4×D + 7×C for F (from eq.2b): 4×C + D + 3×C + D - C = C + D - (6×C + 3×D) + 4×D + 7×C which becomes 6×C + 2×D = C + D - 6×C - 3×D + 4×D + 7×C which becomes 6×C + 2×D = 2×C + 2×D Subtract 2×D and 2×C from both sides of the above equation: 6×C + 2×D - 2×D - 2×C = 2×C + 2×D - 2×D - 2×C which makes 4×C = 0 which means C = 0
Hint #8
Substitute 0 for C in eq.4b: 4×0 + D = A which becomes 0 + D = A which makes D = A
Hint #9
Substitute 0 for C in eq.5a: 3×0 + D = B which becomes 0 + D = B which makes D = B
Hint #10
Substitute 0 for C in eq.3a: E = 6×0 + 3×D which becomes E = 0 + 3×D which makes E = 3×D
Hint #11
Substitute 0 for C in eq.2b: 4×D + 7×0 = F which becomes 4×D + 0 = F which makes 4×D = F
Solution
Substitute D for A and B, 0 for C, 3×D for E, and 4×D for F in eq.1: D + D + 0 + D + 3×D + 4×D = 20 which simplifies to 10×D = 20 Divide both sides of the above equation by 10: 10×D ÷ 10 = 20 ÷ 10 which means D = 2 making A = B = D = 2 E = 3×D = 3 × 2 = 6 F = 4×D = 4 × 2 = 8 and ABCDEF = 220268