Puzzle for March 15, 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.
* AB and BC are 2-digit numbers (not A×B or B×C).
Scratchpad
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Hint #1
eq.4 may be written as: eq.4a) B - D = A - E - C Subtract both D and E from each side of eq.5: A + D - D - E = B - C + E - D - E which becomes eq.5a) A - E = B - C - D
Hint #2
In eq.4a, replace A - E with B - C - D (from eq.5a): B - D = B - C - D - C which becomes B - D = B - 2×C - D Add (D - B) to both sides of the above equation: B - D + (D - B) = B - D - 2×C + (D - B) which simplifies to 0 = -2×C which means 0 = C
Hint #3
Subtract E from each side of eq.2: F - D - E = B + C + E - E which becomes eq.2a) F - D - E = B + C
Hint #4
eq.6 may be expressed as: 10×A + B = 10×B + C - D - E + F which may be written as 10×A + B = 10×B + C + F - D - E In the above equation, replace F - D - E with B + C (from eq.2a): 10×A + B = 10×B + C + B + C which becomes 10×A + B = 11×B + 2×C Substitute 0 for C, and subtract B from each side: 10×A + B - B = 11×B + 2×0 - B which becomes 10×A = 10×B Divide both sides by 10: 10×A ÷ 10 = 10×B ÷ 10 which makes A = B
Hint #5
Substitute A for B, and 0 for C in eq.5: A + D = A - 0 + E Subtract A from both sides of the above equation: A + D - A = A - 0 + E - A which simplifies to D = E
Hint #6
Substitute A for B, 0 for C, and D for E in eq.2: F - D = A + 0 + D Add D to both sides: F - D + D = A + 0 + D + D which becomes eq.2b) F = A + 2×D
Hint #7
Substitute D for E, and A for B in eq.3: D + D + F = A + A which becomes 2×D + F = 2×A In the above equation, substitute A + 2×D for F (from eq.2b): 2×D + A + 2×D = 2×A which becomes A + 4×D = 2×A Subtract A from each side: A + 4×D - A = 2×A - A which makes 4×D = A and which also makes B = A = 4×D
Hint #8
Substitute 4×D for A in eq.2b: F = 4×D + 2×D which makes F = 6×D
Solution
Substitute 4×D for A and B, 0 for C, D for E, and 6×D for F in eq.1: 4×D + 4×D + 0 + D + D + 6×D = 16 which simplifies to 16×D = 16 Divide both sides by 16: 16×D ÷ 16 = 16 ÷ 16 which means D = 1 making A = B = 4×D = 4 × 1 = 4 E = D = 1 F = 6×D = 6 × 1 = 6 and ABCDEF = 440116