Puzzle for December 9, 2021 ( )
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.5, replace B + C with E + F (from eq.4): D + E – B = E + F In the above equation, subtract E from both sides, and add B to both sides: D + E – B – E + B = E + F – E + B which becomes eq.5a) D = F + B
Hint #2
In eq.2, replace D with F + B (from eq.5a): A + B = F + B Subtract B from both sides of the equation above: A + B – B = F + B – B which makes A = F
Hint #3
In eq.3, substitute A + B for D (from eq.2), and A for F: A + B + A = E which becomes eq.3a) 2×A + B = E
Hint #4
In eq.4, substitute 2×A + B for E (from eq.3a), and A for F: B + C = 2×A + B + A which becomes B + C = 3×A + B Subtract B from each side of the above equation: B + C – B = 3×A + B – B which makes C = 3×A
Hint #5
Substitute A + B for D (from eq.2), A for F, 3×A for C, and 2×A + B for E (from eq.3a) in eq.6: B + A + B – A = A + 3×A + 2×A + B + A which becomes 2×B = 7×A + B Subtract B from both sides of the above equation: 2×B – B = 7×A + B – B which makes B = 7×A
Hint #6
Substitute 7×A for B in eq.3a: 2×A + 7×A = E which makes 9×A = E
Hint #7
Substitute 7×A for B in eq.2: A + 7×A = D which makes 8×A = D
Solution
Substitute 7×A for B, 3×A for C, 8×A for D, 9×A for E, and A for F in eq.1: A + 7×A + 3×A + 8×A + 9×A + A = 29 which simplifies to 29×A = 29 Divide both sides of the above equation by 29: 29×A ÷ 29 = 29 ÷ 29 which means A = 1 making B = 7×A = 7 × 1 = 7 C = 3×A = 3 × 1 = 3 D = 8×A = 8 × 1 = 8 E = 9×A = 9 × 1 = 9 F = A = 1 and ABCDEF = 173891