Puzzle for September 22, 2023 ( )
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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.
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Hint #1
Add C to both sides of eq.4: B - E + C = A - C + C which becomes eq.4a) B - E + C = A
Hint #2
In eq.3, substitute (B - E + C) for A (from eq.4a): F - E = E - (B - E + C) which becomes F - E = E - B + E - C which becomes F - E = 2×E - B - C Add E, B, and C to both sides of the above equation: F - E + E + B + C = 2×E - B - C + E + B + C which becomes eq.3a) F + B + C = 3×E
Hint #3
eq.1 may be written as: F + B + C + A + D + E = 35 In the above equation, replace F + B + C with 3×E (from eq.3a), and A + D with E (from eq.2): 3×E + E + E = 35 which makes 5×E = 35 Divide both sides by 5: 5×E ÷ 5 = 35 ÷ 5 which makes E = 7
Hint #4
Add 2×D to both sides of eq.6: A - D + 2×D = C + D + 2×D which becomes A + D = C + 3×D In the equation above, replace A + D with E (from eq.2) : eq.6a) E = C + 3×D
Hint #5
In eq.5, substitute C + 3×D for E (from eq.6a): C - D = C + 3×D - C which becomes C - D = 3×D Add D to both sides of the equation above: C - D + D = 3×D + D which makes C = 4×D
Hint #6
Substitute 7 for E, and 4×D for C in eq.6a: 7 = 4×D + 3×D which makes 7 = 7×D Divide both sides of the above equation by 7: 7 ÷ 7 = 7×D ÷ 7 which makes 1 = D and also makes C = 4×D = 4 × 1 = 4
Hint #7
Substitute 7 for E, and 1 for D in eq.2: 7 = A + 1 Subtract 1 from each side of the equation above: 7 - 1 = A + 1 - 1 which makes 6 = A
Hint #8
Substitute 7 for E, and 6 for A in eq.3: F - 7 = 7 - 6 which becomes F - 7 = 1 Add 7 to both sides of the above equation: F - 7 + 7 = 1 + 7 which makes F = 8
Solution
Substitute 7 for E, 6 for A, and 4 for C in eq.4: B - 7 = 6 - 4 which becomes B - 7 = 2 Add 7 to both sides of the above equation: B - 7 + 7 = 2 + 7 which makes B = 9 and makes ABCDEF = 694178