Puzzle for January 29, 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 positive integer.
Scratchpad
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Hint #1
In eq.5, replace D + F with A + E (from eq.3): A + B + A + E = C + E which becomes 2×A + B + E = C + E Subtract E from each side of the above equation: 2×A + B + E – E = C + E – E which becomes eq.5a) 2×A + B = C
Hint #2
In eq.4, replace C with 2×A + B (from eq.5a): A + 2×A + B = B + D + F which becomes 3×A + B = B + D + F Subtract B from each side of the equation above: 3×A + B – B = B + D + F – B which becomes eq.4a) 3×A = D + F
Hint #3
In eq.2, replace C with 2×A + B (from eq.5a): A + D = B + 2×A + B which becomes A + D = 2×A + 2×B Subtract A from both sides of the equation above: A + D – A = 2×A + 2×B – A which becomes eq.2a) D = A + 2×B
Hint #4
In eq.4a, substitute A + 2×B for D (from eq.2a): 3×A = A + 2×B + F Subtract A and 2×B from both sides of the above equation: 3×A – A – 2×B = A + 2×B + F – A – 2×B which becomes eq.4b) 2×A – 2×B = F
Hint #5
Substitute (2×A + B) for C (from eq.5a), (A + 2×B) for D (from eq.2a), and (2×A – 2×B) for F (from eq.4b) in eq.6: A × (2×A + B) = (A + 2×B) × (2×A – 2×B) which becomes 2×A² + A×B = 2×A² – 2×A×B + 4×A×B – 4×B² which becomes 2×A² + A×B = 2×A² + 2×A×B – 4×B² Subtract 2×A² and A×B from each side of the equation above: 2×A² + A×B – 2×A² – A×B = 2×A² + 2×A×B – 4×B² – 2×A² – A×B which becomes 0 = A×B – 4×B² Add 4×B² to both sides: 0 + 4×B² = A×B – 4×B² + 4×B² which makes 4×B² = A×B Divide both sides by B: 4×B² ÷ B = A×B ÷ B which makes 4×B = A
Hint #6
Substitute (4×B) for A in eq.4b: 2×(4×B) – 2×B = F which becomes 8×B – 2×B = F which makes 6×B = F
Hint #7
Substitute 4×B for A in eq.2a: D = 4×B + 2×B which makes D = 6×B
Hint #8
Substitute (4×B) for A in eq.5a: 2×(4×B) + B = C which becomes 8×B + B = C which makes 9×B = C
Hint #9
Substitute 4×B for A, and 6×B for D and F in eq.3: 4×B + E = 6×B + 6×B which becomes 4×B + E = 12×B Subtract 4×B from each side of the equation above: 4×B + E - 4×B = 12×B - 4×B which makes E = 8×B
Solution
Substitute 4×B for A, 9×B for C, 6×B for D and F, and 8×B for E in eq.1: 4×B + B + 9×B + 6×B + 8×B + 6×B = 34 which simplifies to 34×B = 34 Divide both sides of the above equation by 34: 34×B ÷ 34 = 34 ÷ 34 which means B = 1 making A = 4×B = 4×1 = 4 C = 9×B = 9×1 = 9 D = F = 6×B = 6×1 = 6 E = 8×B = 8×1 = 8 and ABCDEF = 419686