Puzzle for May 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.6, replace A + C with E (from eq.3): B + D + F = E + E which becomes eq.6a) B + D + F = 2×E
Hint #2
eq.1 may be written as: A + C + E + B + D + F = 28 In the above equation, replace A + C with E (from eq.3), and B + D + F with 2×E (from eq.6a): E + E + 2×E = 28 which becomes 4×E = 28 Divide both sides by 4: 4×E ÷ 4 = 28 ÷ 4 which makes E = 7
Hint #3
In eq.3, substitute 7 for E: 7 = A + C Subtract C from each side of the equation above: 7 - C = A + C - C which becomes eq.3a) 7 - C = A
Hint #4
Substitute 7 - C for A (from eq.3a), and 7 for E in eq.4: 7 - C + B = C + 7 In the above equation, subtract 7 from both sides, and add C to both sides: 7 - C + B - 7 + C = C + 7 - 7 + C which makes B = 2×C
Hint #5
Substitute 7 - C for A (from eq.3a) into eq.2: C = 7 - C + F In the equation above, subtract 7 from both sides, and add C to both sides: C - 7 + C = 7 - C + F - 7 + C which becomes eq.2a) 2×C - 7 = F
Hint #6
Substitute 2×C for B, 2×C - 7 for F (from eq.2a), and 7 for E in eq.6a: 2×C + D + 2×C - 7 = 2×7 which becomes 4×C + D - 7 = 14 In the above equation, subtract 4×C from both sides, and add 7 to both sides: 4×C + D - 7 - 4×C + 7 = 14 - 4×C + 7 which becomes eq.6b) D = 21 - 4×C
Hint #7
Substitute 21 - 4×C for D (from eq.6b), 7 for E, and 2×C for B in eq.5: 21 - 4×C + 7 = 2×C + C which becomes 28 - 4×C = 3×C Add 4×C to both sides of the above equation: 28 - 4×C + 4×C = 3×C + 4×C which becomes 28 = 7×C Divide both sides by 7: 28 ÷ 7 = 7×C ÷ 7 which makes 4 = C
Solution
Since C = 4, then: A = 7 - C = 7 - 4 = 3 (from eq.3a) B = 2×C = 2×4 = 8 D = 21 - 4×C = 21 - 4×4 = 21 - 16 = 5 (from eq.6b) F = 2×C - 7 = 2×4 - 7 = 8 - 7 = 1 (from eq.2a) and ABCDEF = 384571