Puzzle for November 18, 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
eq.5 may be written as: A + D – C = B + C In the equation above, replace A + D with B + F (from eq.2): B + F – C = B + C Subtract B from both sides, and add C to both sides: B + F – C – B + C = B + C – B + C which makes F = 2×C
Hint #2
eq.6 may be written as: B = (A + C + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × B = 4 × (A + C + E + F) ÷ 4 which becomes 4×B = A + C + E + F which may be written as eq.6a) 4×B = A + E + C + F
Hint #3
In eq.6a, replace A + E with C + F (from eq.3): 4×B = C + F + C + F which becomes 4×B = 2×C + 2×F Replace F with (2×C) in the above equation: 4×B = 2×C + 2×(2×C) which is equivalent to 4×B = 2×C + 4×C which becomes 4×B = 6×C Divide both sides by 4: 4×B ÷ 4 = 6×C ÷ 4 which makes B = 1½×C
Hint #4
In eq.4, substitute 1½×C for B: C + E = A + 1½×C Subtract C from each side of the equation above: C + E – C = A + 1½×C – C which becomes eq.4a) E = A + ½×C
Hint #5
Substitute 1½×C for B, A + ½×C for E (from eq.4a), and 2×C for F in eq.6a: 4×(1½×C) = A + A + ½×C + C + 2×C which becomes 6×C = 2×A + 3½×C Subtract 3½×C from both sides of the equation above: 6×C – 3½×C = 2×A + 3½×C – 3½×C which makes 2½×C = 2×A Divide both sides by 2: 2½×C ÷ 2 = 2×A ÷ 2 which makes 1¼×C = A
Hint #6
Substitute 1¼×C for A in eq.4a: E = 1¼×C + ½×C which makes E = 1¾×C
Hint #7
Substitute 1½×C for B, 2×C for F, and 1¼×C for A in eq.2: 1½×C + 2×C = 1¼×C + D which becomes 3½×C = 1¼×C + D Subtract 1¼×C from both sides of the above equation: 3½×C – 1¼×C = 1¼×C + D – 1¼×C which makes 2¼×C = D
Solution
Substitute 1¼×C for A, 1½×C for B, 2¼×C for D, 1¾×C for E, and 2×C for F in eq.1: 1¼×C + 1½×C + C + 2¼×C + 1¾×C + 2×C = 39 which simplifies to 9¾×C = 39 Divide both sides of the above equation by 9¾: 9¾×C ÷ 9¾ = 39 ÷ 9¾ which means C = 4 making A = 1¼×C = 1¼ × 4 = 5 B = 1½×C = 1½ × 4 = 6 D = 2¼×C = 2¼ × 4 = 9 E = 1¾×C = 1¾ × 4 = 7 F = 2×C = 2 × 4 = 8 and ABCDEF = 564978