Puzzle for November 21, 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
Add C to both sides of eq.5: A + C + C = F – C + C which becomes eq.5a) A + 2×C = F In eq.3, replace F with A + 2×C (from eq.5a): A + 2×C – A = A + C – D which becomes 2×C = A + C – D Subtract C from both sides of the above equation: 2×C – C = A + C – D – C which becomes eq.3a) C = A – D
Hint #2
In eq.2, replace C with A – D (from eq.3a): A – D + D + E = A + B which becomes A + E = A + B Subtract A from both sides of the above equation: A + E – A = A + B – A which makes B = E
Hint #3
In eq.6, substitute B for E: B + C = A + D + B Subtract B from both sides of the above equation: B + C – B = A + D + B – B which becomes eq.6a) C = A + D
Hint #4
Substitute A + D for C (from eq.6a) in eq.3a: A + D = A – D Subtract A from both sides of the above equation: A + D – A = A – D – A which makes D = –D which means D = 0
Hint #5
Substitute 0 for D in eq.3a: C = A – 0 which means C = A
Hint #6
Substitute A for C in eq.5a: A + 2×A = F which makes 3×A = F
Hint #7
Substitute A for C, 3×A for F, and B for E in eq.4: B – A – 3×A = 3×A – B which becomes B – 4×A = 3×A – B Add B and 4×A to both sides of the above equation: B – 4×A + B + 4×A = 3×A – B + B + 4×A 2×B = 7×A Divide both sides of the above equation by 2: 2×B ÷ 2 = 7×A ÷ 2 which makes B = 3½×A and also makes E = B = 3½×A
Solution
Substitute 3½×A for B and E, A for C, 0 for D, and 3×A for F in eq.1: A + 3½×A + A + 0 + 3½×A + 3×A = 24 which simplifies to 12×A = 24 Divide both sides of the above equation by 12: 12×A ÷ 12 = 24 ÷ 12 which means A = 2 making B = E = 3½×A = 3½ × 2 = 7 C = A = 2 F = 3×A = 3 × 2 = 6 and ABCDEF = 272076