Puzzle for October 1, 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.5, replace B with A + F (from eq.3): A + E = A + F + F which becomes A + E = A + 2×F Subtract A from both sides of the above equation: A + E – A = A + 2×F – A which makes eq.5a) E = 2×F
Hint #2
In eq.2, replace E with A + C + D (from eq.6): D + A + C + D = B + C which becomes A + C + 2×D = B + C Subtract C from both sides of the above equation: A + C + 2×D – C = B + C – C which becomes eq.2a) A + 2×D = B
Hint #3
In eq.2a, substitute A + F for B (from eq.3): A + 2×D = A + F Subtract A from each side of the equation above: A + 2×D – A = A + F – A which becomes 2×D = F
Hint #4
Substitute (2×D) for F in eq.5a: E = 2×(2×D) which makes E = 4×D
Hint #5
In eq.6, substitute 4×D for E, and A – D for C + D (from eq.4): 4×D = A + A – D which becomes 4×D = 2×A – D Add D to both sides of the equation above: 4×D + D = 2×A – D + D which means 5×D = 2×A Divide both sides by 2: 5×D ÷ 2 = 2×A ÷ 2 which makes 2½×D = A
Hint #6
Substitute 2½×D for A, and 2×D for F in eq.3: B = 2½×D + 2×D which makes B = 4½×D
Hint #7
Substitute 2½×D for A in eq.4: C + D = 2½×D – D which becomes C + D = 1½×D Subtract D from each side of the above equation: C + D – D = 1½×D – D which makes C = ½×D
Solution
Substitute 2½×D for A, 4½×D for B, ½×D for C, 4×D for E, and 2×D for F in eq.1: 2½×D + 4½×D + ½×D + D + 4×D + 2×D = 29 which simplifies to 14½×D = 29 Divide both sides of the equation above by 14½: 14½×D ÷ 14½ = 29 ÷ 14½ which means D = 2 making A = 2½×D = 2½ × 2 = 5 B = 4½×D = 4½ × 2 = 9 C = ½×D = ½ × 2 = 1 E = 4×D = 4 × 2 = 8 F = 2×D = 2 × 2 = 4 and ABCDEF = 591284