Puzzle for March 3, 2023 ( )
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 B and D to both sides of eq.6: F - B + B + D = B - D + B + D which becomes eq.6a) F + D = 2×B Add C and B to both sides of eq.2: B - C + C + B = E - B + C + B which becomes eq.2a) 2×B = E + C
Hint #2
In eq.2a, replace 2×B with F + D (from eq.6a): F + D = E + C Subtract C from each side of the equation above: F + D - C = E + C - C which becomes eq.2b) F + D - C = E
Hint #3
In eq.4, replace E with F + D - C (from eq.2b): D + F + D - C = A + F which becomes 2×D + F - C = A + F In the equation above, subtract F from both sides, and add C to both sides: 2×D + F - C - F + C = A + F - F + C which becomes eq.2c) 2×D = A + C
Hint #4
In eq.5, substitute 2×D for A + C (from eq.2c): E - A = 2×D - D which becomes E - A = D Add A to both sides of the equation above: E - A + A = D + A which becomes eq.5a) E = D + A
Hint #5
Substitute D + A for E (from eq.5a) in eq.4: D + D + A = A + F which becomes 2×D + A = A + F Subtract A from each side of the above equation: 2×D + A - A = A + F - A which makes 2×D = F
Hint #6
Substitute 2×D for F in eq.6a: 2×D + D = 2×B which makes 3×D = 2×B Divide both sides of the above equation by 2: 3×D ÷ 2 = 2×B ÷ 2 which makes 1½×D = B
Hint #7
In eq.3, substitute D + A for E (from eq.5a): C + D + A = A - C + D In the above equation, subtract D and A from both sides, and add C to both sides: C + D + A - D - A + C = A - C + D - D - A + C which simplifies to 2×C = 0 which means C = 0
Hint #8
Substitute 0 for C in eq.2c: 2×D = A + 0 which makes 2×D = A
Hint #9
Substitute 2×D for A in eq.5a: E = D + 2×D which makes E = 3×D
Solution
Substitute 2×D for A and F, 1½×D for B, 0 for C, and 3×D for E in eq.1: 2×D + 1½×D + 0 + D + 3×D + 2×D = 19 which simplifies to 9½×D = 19 Divide both sides of the above equation by 9½: 9½×D ÷ 9½ = 19 ÷ 9½ which means D = 2 making A = F = 2×D = 2 × 2 = 4 B = 1½×D = 1½ × 2 = 3 E = 3×D = 3 × 2 = 6 and ABCDEF = 430264