Puzzle for August 12, 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
Add B and E to both sides of eq.5: A – B + E + B + E = B + C – E + B + E which becomes eq.5a) A + 2×E = 2×B + C Add B and E to both sides of eq.4: B – E + B + E = A – B + B + E which becomes eq.4a) 2×B = A + E
Hint #2
In eq.5a, replace 2×B with A + E (from eq.4a): A + 2×E = A + E + C Subtract A and E from both sides of the equation above: A + 2×E – A – E = A + E + C – A – E which simplifies to E = C
Hint #3
In eq.3, replace E with C: C + F = C Subtract C from both sides of the equation above: C + F – C = C – C which makes F = 0
Hint #4
In eq.6, substitute C for E, and 0 for F: C – D + C = A + D – 0 which becomes 2×C – D = A + D Add D to both sides of the above equation: 2×C – D + D = A + D + D which becomes eq.6a) 2×C = A + 2×D
Hint #5
Substitute C + D for A (from eq.2) in eq.6a: 2×C = C + D + 2×D which becomes 2×C = C + 3×D Subtract C from each side of the above equation: 2×C – C = C + 3×D – C which makes C = 3×D and also makes E = C = 3×D
Hint #6
Substitute 3×D for C in eq.2: 3×D + D = A which makes 4×D = A
Hint #7
Substitute 3×D for E, and 4×D for A in eq.4: B – 3×D = 4×D – B Add 3×D and B to both sides of the above equation: B – 3×D + 3×D + B = 4×D – B + 3×D + B which makes 2×B = 7×D Divide both sides by 2: 2×B ÷ 2 = 7×D ÷ 2 which makes B = 3½×D
Solution
Substitute 4×D for A, 3½×D for B, 3×D for C and E, and 0 for F in eq.1: 4×D + 3½×D + 3×D + D + 3×D + 0 = 29 which simplifies to 14½×D = 29 Divide both sides of the above equation by 14½: 14½×D ÷ 14½ = 29 ÷ 14½ which means D = 2 making A = 4×D = 4 × 2 = 8 B = 3½×D = 3½ × 2 = 7 C = E = 3×D = 3 × 2 = 6 and ABCDEF = 876260