Puzzle for December 25, 2019 ( )
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 A with B – E (from eq.2): B + D – E = B – E + F In the above equation, subtract B from both sides, and add E to each side: B + D – E – B + E = B – E + F – B + E which simplifies to D = F
Hint #2
Add E to both sides of eq.2: B – E + E = A + E which becomes B = A + E In eq.3, replace F with D, and B with A + E: E + D = A + E + C Subtract E from both sides of the above equation: A + E + C – E = E + D – E which becomes eq.3a) A + C = D
Hint #3
Substitute A + C for D (from eq.3a) in eq.6: C + E – A = A + A + C In the above equation, add A to both sides, and subtract C from each side:: C + E – C – A + A = A + A + C + A – C which makes E = 3×A
Hint #4
Substitute 3×A for E in eq.2: B – 3×A = A Add 3×A to both sides of the equation above: B – 3×A + 3×A = A + 3×A which makes B = 4×A
Hint #5
Substitute A + C for D (from eq.3a), 4×A for B, and 3×A for E in eq.4: A + C + A + C = 4×A + 3×A which becomes 2×A + 2×C = 7×A Subtract 2×A from both sides of the equation above: 2×A + 2×C – 2×A = 7×A – 2×A which makes 2×C = 5×A Divide both sides by 2: 2×C ÷ 2 = 5×A ÷ 2 which means C = 2½×A
Hint #6
Substitute 2½×A for C in eq.3a: A + 2½×A = D which makes 3½×A = D and also makes F = D = 3½×A
Solution
Substitute 4×A for B, 2½×A for C, 3½×A for D and F, and 3×A for E in eq.1: A + 4×A + 2½×A + 3½×A + 3×A + 3½×A = 35 which simplifies to 17½×A = 35 Divide both sides of the above equation by 17½: 17½×A ÷ 17½ = 35 ÷ 17½ which means A = 2 making B = 4×A = 4 × 2 = 8 C = 2½×A = 2½ × 2 = 5 D = F = 3½×A = 3½ × 2 = 7 E = 3×A = 3 × 2 = 6 and ABCDEF = 285767