Puzzle for November 25, 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 C and F to both sides of eq.3: F – C + C + F = A – F + C + F which becomes eq.3a) 2×F = A + C Subtract F from both sides of eq.5: E + F – F = A + B + C – F – F which becomes E = A + B + C – 2×F which may be written as eq.5a) E = A + C + B – 2×F
Hint #2
In eq.5a, replace A + C with 2×F (from eq.3a): E = 2×F + B – 2×F which makes E = B
Hint #3
In eq.2, replace E with B: D – B = B – D Add B and D to both sides of the above equation: D – B + B + D = B – D + B + D which makes 2×D = 2×B Divide each side by 2: 2×D ÷ 2 = 2×B ÷ 2 which makes D = B and also makes E = B = D
Hint #4
Subtract C from both sides of eq.3a: 2×F – C = A + C – C which becomes eq.3b) 2×F – C = A In eq.4, replace A with 2×F – C (from eq.3b): C + E = 2×F – C – C + F which becomes C + E = 3×F – 2×C Subtract C from each side of the above equation: C + E – C = 3×F – 2×C – C which makes E = 3×F – 3×C and also makes eq.4a) B = D = E = 3×F – 3×C
Hint #5
In eq.6, substitute 3×F – 3×C for B, E, and D (from eq.4a): 3×F – 3×C – C + 3×F – 3×C = C + 3×F – 3×C + F which becomes 6×F – 7×C = 4×F – 2×C In the equation above, add 7×C to both sides, and subtract 4×F from both sides: 6×F – 7×C + 7×C – 4×F = 4×F – 2×C + 7×C – 4×F which becomes 2×F = 5×C Divide both sides by 2: 2×F ÷ 2 = 5×C ÷ 2 which makes F = 2½×C
Hint #6
Substitute (2½×C) for F in eq.3b: 2×(2½×C) – C = A which becomes 5×C – C = A which makes 4×C = A
Hint #7
Substitute (2½×C) for F in eq.4a: B = D = E = 3×(2½×C) – 3×C which becomes B = D = E = 7½×C – 3×C which makes B = D = E = 4½×C
Solution
Substitute 4×C for A, 4½×C for B and D and E, and 2½×C for F in eq.1: 4×C + 4½×C + C + 4½×C + 4½×C + 2½×C = 42 which simplifies to 21×C = 42 Divide both sides of the equation above by 21: 21×C ÷ 21 = 42 ÷ 21 which means C = 2 making A = 4×C = 4 × 2 = 8 B = D = E = 4½×C = 4½ × 2 = 9 F = 2½×C = 2½ × 2 = 5 and ABCDEF = 892995