Puzzle for April 27, 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.2, replace E with D – F (from eq.4): B = D + D – F + F which makes eq.2a) B = 2×D
Hint #2
In eq.3, replace B with 2×D: D + C = 2×D – C In the equation above, add C to both sides, and subtract D from both sides: D + C + C – D = 2×D – C + C – D which makes 2×C = D
Hint #3
In eq.2a, substitute (2×C) for D: B = 2×(2×C) which makes B = 4×C
Hint #4
Substitute 4×C for B in eq.6: C + F = 4×C – C which becomes C + F = 3×C Subtract C from both sides of the equation above: C + F – C = 3×C – C which makes F = 2×C
Hint #5
Substitute 2×C for both D and F in eq.4: E = 2×C – 2×C which makes E = 0
Hint #6
Substitute 4×C for B, and 2×C for D in eq.5: 4×C + 2×C – A = A – C – 2×C which becomes 6×C – A = A – 3×C Add A and 3×C to each side of the equation above: 6×C – A + A + 3×C = A – 3×C + A + 3×C which becomes 9×C = 2×A Divide both sides of the above equation by 2: 9×C ÷ 2 = 2×A ÷ 2 which makes 4½×C = A
Solution
Substitute 4½×C for A, 4×C for B, 2×C for D and F, and 0 for E in eq.1: 4½×C + 4×C + C + 2×C + 0 + 2×C = 27 which simplifies to 13½×C = 27 Divide both sides of the equation above by 13½: 13½×C ÷ 13½ = 27 ÷ 13½ = 2 which means C = 2 making A = 4½×C = 4½ × 2 = 9 B = 4×C = 4 × 2 = 8 D = F = 2×C = 2 × 2 = 4 and ABCDEF = 982404