Puzzle for August 23, 2019 ( )
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.6, replace C + D with B + F (from eq.3): E + F = B + B + F which becomes E + F = 2×B + F Subtract F from both sides of the equation above: E + F - F = 2×B + F - F which makes E = 2×B
Hint #2
In eq.6, replace E with C - D (from eq.5): C - D + F = B + C + D In the above equation, subtract C from both sides, and add D to both sides: C - D + F - C + D = B + C + D - C + D which becomes eq.6a) F = B + 2×D
Hint #3
In eq.2, substitute B + 2×D for F (from eq.6a): D + B + 2×D = A + B Subtract B from each side of the equation above: D + B + 2×D - B = A + B - B which means 3×D = A
Hint #4
Substitute 3×D for A, and 2×B for E in eq.4: 3×D + C = B + 2×B Subtract 3×D from both sides of the above equation: 3×D + C - 3×D = B + 2×B - 3×D which becomes eq.4a) C = 3×B - 3×D
Hint #5
Substitute B + 2×D for F (from eq.6a) in eq.3: B + B + 2×D = C + D which becomes eq.3a) 2×B + D = C
Hint #6
Substitute 3×B - 3×D for C (from eq.4a) in eq.3a: 2×B + D = 3×B - 3×D In the above equation, add 3×D to both sides, and subtract 2×B from each side: 2×B + D + 3×D - 2×B = 3×B - 3×D + 3×D - 2×B which simplifies to 4×D = B
Hint #7
Substitute (4×D) for B in eq.3a: 2×(4×D) + D = C which means 9×D = C
Hint #8
Substitute 9×D for C in eq.5: 9×D - D = E which means 8×D = E
Hint #9
Substitute 4×D for B in eq.6a: F = 4×D + 2×D which makes F = 6×D
Solution
Substitute 3×D for A, 4×D for B, 9×D for C, 8×D for E, and 6×D for F in eq.1: 3×D + 4×D + 9×D + D + 8×D + 6×D = 31 which becomes 31×D = 31 Divide both sides of the equation above by 31: 31×D ÷ 31 = 31 ÷ 31 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 4×D = 4 × 1 = 4 C = 9×D = 9 × 1 = 9 E = 8×D = 8 × 1 = 8 F = 6×D = 6 × 1 = 6 and ABCDEF = 349186