Puzzle for November 14, 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
Add F and E to each side of eq.6: A – F + F + E = C – E + F + E which becomes A + E = C + F In eq.5, replace C + F with A + E: B + E = A + E Subtract E from both sides of the equation above: B + E – E = A + E – E which makes B = A
Hint #2
In eq.2, replace B with A: E = A + A which makes E = 2×A
Hint #3
In eq.4, substitute A for B, and 2×A for E: C + D = A + 2×A + F which becomes eq.4a) C + D = 3×A + F
Hint #4
Substitute D + F for C (from eq.3) into eq.4a: D + F + D = 3×A + F which becomes 2×D + F = 3×A + F Subtract F from both sides of the above equation: 2×D + F – F = 3×A + F – F which means 2×D = 3×A Divide both sides by 2: 2×D ÷ 2 = 3×A ÷ 2 which makes D = 1½×A
Hint #5
Substitute 1½×A for D in eq.3: eq.3a) 1½×A + F = C Substitute A for B, and 2×A for E in eq.5: A + 2×A = C + F which becomes eq.5a) 3×A = C + F
Hint #6
Substitute 1½×A + F for C (from eq.3a) in eq.5a: 3×A = 1½×A + F + F which becomes 3×A = 1½×A + 2×F Subtract 1½×A from each side of the above equation: 3×A – 1½×A = 1½×A + 2×F – 1½×A which makes 1½×A = 2×F Divide both sides by 2: 1½×A ÷ 2 = 2×F ÷ 2 which means ¾×A = F
Hint #7
Substitute ¾×A for F in eq.3a: 1½×A + ¾×A = C which makes 2¼×A = C
Solution
Substitute A for B, 2¼×A for C, 1½×A for D, 2×A for E, and ¾×A for F in eq.1: A + A + 2¼×A + 1½×A + 2×A + ¾×A = 34 which simplifies to 8½×A = 34 Divide both sides of the equation above by 8½: 8½×A ÷ 8½ = 34 ÷ 8½ which means A = 4 making B = A = 4 C = 2¼×A = 2¼ × 4 = 9 D = 1½×A = 1½ × 4 = 6 E = 2×A = 2 × 4 = 8 F = ¾×A = ¾ × 4 = 3 and ABCDEF = 449683