Puzzle for October 26, 2020 ( )
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 F with A + B (from eq.2), and A + D with E (from eq.5): A + B – A = E which makes B = E
Hint #2
In eq.4, replace E with B: D + B = B – D + F In the above equation, subtract B from both sides, and add D to both sides: D + B – B + D = B – D + F – B + D which makes eq.4a) 2×D = F
Hint #3
In eq.6, substitute 2×D for F: 2×D – A = A + D In the above equation, add A to both sides, and subtract D from both sides: 2×D – A + A – D = A + D + A – D which makes D = 2×A
Hint #4
Substitute (2×A) for D in eq.4a: 2×(2×A) = F which makes 4×A = F
Hint #5
Substitute 2×A for D in eq.5: E = A + 2×A which makes E = 3×A and also makes B = E = 3×A
Hint #6
Substitute 3×A for B and E, and 4×A for F in eq.3: 3×A – C + 3×A + 4×A = A + C which becomes 10×A – C = A + C In the above equation, add C to both sides, and subtract A from both sides: 10×A – C + C – A = A + C + C – A which makes 9×A = 2×C Divide both sides by 2: 9×A ÷ 2 = 2×C ÷ 2 which makes 4½×A = C
Solution
Substitute 3×A for B and E, 4½×A for C, 2×A for D, and 4×A for F in eq.1: A + 3×A + 4½×A + 2×A + 3×A + 4×A = 35 which simplifies to 17½×A = 35 Divide both sides of the equation above by 17½: 17½×A ÷ 17½ = 35 ÷ 17½ which means A = 2 making B = E = 3×A = 3 × 2 = 6 C = 4½×A = 4½ × 2 = 9 D = 2×A = 2 × 2 = 4 F = 4×A = 4 × 2 = 8 and ABCDEF = 269468