Puzzle for September 27, 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 D and B to both sides of eq.5: B - D + D + B = E + F - B + D + B which becomes eq.5a) 2×B = E + F + D
Hint #2
In eq.5a, replace E + F with B + C (from eq.3): 2×B = B + C + D Subtract B from each side of the above equation: 2×B - B = B + C + D - B which becomes eq.5b) B = C + D
Hint #3
In eq.3, replace B with C + D (from eq.5b): E + F = C + D + C which becomes eq.3a) E + F = 2×C + D
Hint #4
In eq.6, substitute 2×C + D for E + F (from eq.3a): C + 2×C + D = A + D which becomes 3×C + D = A + D Subtract D from both sides of the equation above: 3×C + D - D = A + D - D which makes 3×C = A
Hint #5
Substitute 3×C for A in eq.4: 3×C - D = D - C Add D and C to both sides of the above equation: 3×C - D + D + C = D - C + D + C which makes 4×C = 2×D Divide both sides by 2: 4×C ÷ 2 = 2×D ÷ 2 which makes 2×C = D
Hint #6
Substitute 2×C for D in eq.5b: B = C + 2×C which makes B = 3×C
Hint #7
Substitute 2×C for D, and 3×C for A and B in eq.2: 2×C + F = 3×C + 3×C which becomes 2×C + F = 6×C Subtract 2×C from each side of the above equation: 2×C + F - 2×C = 6×C - 2×C which makes F = 4×C
Hint #8
Substitute 4×C for F, and 3×C for B in eq.3: E + 4×C = 3×C + C which becomes E + 4×C = 4×C Subtract 4×C from each side of the equation above: E + 4×C - 4×C = 4×C - 4×C which makes E = 0
Solution
Substitute 3×C for A and B, 2×C for D, 0 for E, and 4×C for F in eq.1: 3×C + 3×C + C + 2×C + 0 + 4×C = 13 which simplifies to 13×C = 13 Divide both sides of the above equation by 13: 13×C ÷ 13 = 13 ÷ 13 which means C = 1 making A = B = 3×C = 3 × 1 = 3 D = 2×C = 2 × 1 = 2 F = 4×C = 4 × 1 = 4 and ABCDEF = 331204