Puzzle for June 10, 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.2, replace B with F - D (from eq.5): E + F = F - D + D which becomes E + F = F Subtract F from both sides of the equation above: E + F - F = F - F which makes E = 0
Hint #2
In eq.4, replace E with 0: C - 0 = F - C which becomes C = F - C Add C to both sides of the above equation: C + C = F - C + C which makes 2×C = F
Hint #3
Add C to both sides of eq.3: B - C + C = C + D + C which becomes B = 2×C + D In the above equation, substitute F for 2×C, and substitute F - D for B (from eq.5): F - D = F + D Add D to both sides, and subtract F from both sides: F - D + D - F = F + D + D - F which becomes 0 = 2×D which means 0 = D
Hint #4
Substitute 0 for D in eq.5: F - 0 = B which makes F = B and which means B = F = 2×C
Hint #5
Add A and C to both sides of eq.6: A - C + A + C = B - A + A + C which becomes 2×A = B + C Substitute 2×C for B in the above equation: 2×A = 2×C + C which becomes 2×A = 3×C Divide both sides by 2: 2×A ÷ 2 = 3×C ÷ 2 which makes A = 1½×C
Solution
Substitute 1½×C for A, 2×C for B and F, and 0 for D and E in eq.1: 1½×C + 2×C + C + 0 + 0 + 2×C = 26 which simplifies to 6½×C = 26 Divide both sides by 6½: 6½×C ÷ 6½ = 26 ÷ 6½ which means C = 4 making B = F = 2×C = 2 × 4 = 8 A = 1½C = 1½ × 4 = 6 and ABCDEF = 684008