Puzzle for August 18, 2023 ( )
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 A, B, and D to both sides of eq.3: F - A + A + B + D = A - B - D + A + B + D which becomes eq.3a) F + B + D = 2×A Add A to both sides of eq.2: C + F + A = A + B + A which becomes eq.2a) C + F + A = 2×A + B
Hint #2
In eq.2a, replace 2×A with F + B + D (from eq.3a): C + F + A = F + B + D + B which becomes C + F + A = F + 2×B + D Subtract F from each side of the above equation: C + F + A - F = F + 2×B + D - F which becomes C + A = 2×B + D which is the same as eq.2b) A + C = 2×B + D
Hint #3
eq.6 may be written as: B + D = (A + C + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × (B + D) = 3 × (A + C + F) ÷ 3 which becomes eq.6a) 3×B + 3×D = A + C + F
Hint #4
In eq.6a, replace A + C with 2×B + D (from eq.2b): 3×B + 3×D = 2×B + D + F Subtract 2×B and D from both sides of the equation above: 3×B + 3×D - 2×B - D = 2×B + D + F - 2×B - D which becomes eq.6b) B + 2×D = F
Hint #5
In eq.4, substitute B + 2×D for F (from eq.6b): E + B + 2×D = B + C + D Subtract B and D from each side of the equation above: E + B + 2×D - B - D = B + C + D - B - D which simplifies to eq.4a) E + D = C
Hint #6
Substitute (E + D) for C (from eq.4a) in eq.5: A - (E + D) + E = (E + D) + D - E which becomes A - E - D + E = 2×D which becomes A - D = 2×D Add D to both sides of the above equation: A - D + D = 2×D + D which makes A = 3×D
Hint #7
Substitute B + 2×D for F (from eq.6b), and (3×D) for A in eq.3a: B + 2×D + B + D = 2×(3×D) which becomes 2×B + 3×D = 6×D Subtract 3×D from each side of the above equation: 2×B + 3×D - 3×D = 6×D - 3×D which makes 2×B = 3×D Divide both sides by 2: 2×B ÷ 2 = 3×D ÷ 2 which makes B = 1½×D
Hint #8
Substitute 1½×D for B in eq.6b: 1½×D + 2×D = F which makes 3½×D = F
Hint #9
Substitute 3½×D for F, 3×D for A, and 1½×D for B in eq.2: C + 3½×D = 3×D + 1½×D which becomes C + 3½×D = 4½×D Subtract 3½×D from both sides of the above equation: C + 3½×D - 3½×D = 4½×D - 3½×D which makes C = D
Hint #10
Substitute C for D in eq.4a: E + C = C Subtract C from each side of the above equation: E + C - C = C - C which makes E = 0
Solution
Substitute 3×D for A, 1½×D for B, D for C, 0 for E, and 3½×D for F in eq.1: 3×D + 1½×D + D + D + 0 + 3½×D = 20 which simplifies to 10×D = 20 Divide both sides of the above equation by 10: 10×D ÷ 10 = 20 ÷ 10 which means D = 2 making A = 3×D = 3 × 2 = 6 B = 1½×D = 1½ × 2 = 3 C = D = 2 F = 3½×D = 3½ × 2 = 7 and ABCDEF = 632207