Puzzle for April 2, 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 D to both sides of eq.2: F + D = A - D + D which becomes F + D = A In eq.3, replace A with F + D: F + D + D = C + F Subtract F from each side of the above equation: F + D + D - F = C + F - F which means 2×D = C
Hint #2
In eq.5, substitute 2×D for C: B + D - F = 2×D + E Subtract D from both sides: B + D - F - D = 2×D + E - D which becomes B - F = D + E In the above equation, replace D + E with B + F (from eq.4): B - F = B + F Add (F - B) to both sides of the above equation: B - F + (F - B) = B + F + (F - B) which simplifies to 0 = 2×F which means 0 = F
Hint #3
In eq.2, substitute 0 for F: 0 = A - D Add D to each side of the above equation: 0 + D = A - D + D which makes D = A
Hint #4
Substitute 2×D for C, and D for A in eq.6: 2×D - B = D ÷ D which becomes 2×D - B = 1 (assumes D ≠ 0) Add (B - 1) to each side of the equation above: 2×D - B + (B - 1) = 1 + (B - 1) which becomes eq.6a) 2×D - 1 = B
Hint #5
Substitute 2×D - 1 for B (from eq.6a), and 0 for F in eq.4: D + E = 2×D - 1 + 0 Subtract D from each side: D + E - D = 2×D - 1 + 0 - D which makes eq.4a) E = D - 1
Solution
Substitute D for A, 2×D - 1 for B (from eq.6a), 2×D for C, D - 1 for E (from eq.4a), and 0 for F in eq.1: D + 2×D - 1 + 2×D + D + D - 1 + 0 = 26 which simplifies to 7×D - 2 = 26 Add 2 to each side of the equation above: 7×D - 2 + 2 = 26 + 2 which makes 7×D = 28 Divide both sides by 7: 7×D ÷ 7 = 28 ÷ 7 which makes D = 4 making A = D = 4 B = 2×D - 1 = 2×4 - 1 = 8 - 1 = 7 (from eq.6a) C = 2×D = 2×4 = 8 E = D - 1 = 4 - 1 = 3 (from eq.4a) and ABCDEF = 478430