Puzzle for September 9, 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 F to both sides of eq.2: B + F + F = C - F + F which becomes eq.2a) B + 2×F = C
Hint #2
In eq.3, replace C with B + 2×F (from eq.2a): B + 2×F + D = A + B Subtract B from each side of the equation above: B + 2×F + D - B = A + B - B which becomes eq.3a) 2×F + D = A
Hint #3
In eq.4, replace A with 2×F + D (from eq.3a): D + E - (2×F + D + F) = 2×F + D - D + F which becomes D + E - 2×F - D - F = 3×F which becomes E - 3×F = 3×F Add 3×F to both sides of the above equation: E - 3×F + 3×F = 3×F + 3×F which becomes E = 6×F
Hint #4
In eq.5, substitute 6×F for E: 6×F = B + D Subtract D from each side of the above equation: 6×F - D = B + D - D which becomes eq.5a) 6×F - D = B
Hint #5
Substitute 6×F - D for B (from eq.5a) in eq.2a: 6×F - D + 2×F = C which becomes eq.2b) 8×F - D = C
Hint #6
Substitute 2×F + D for A (from eq.3a), 6×F - D for B (from eq.5a), 8×F - D for C (from eq.2a), and 6×F for E in eq.1: 2×F + D + 6×F - D + 8×F - D + D + 6×F + F = 23 which simplifies to 23×F = 23 Divide both sides of the above equation by 23: 23×F ÷ 23 = 23 ÷ 23 which means F = 1 making E = 6×F = 6 × F = 6
Hint #7
Substitute 1 for F in eq.5a: 6×1 - D = B which becomes eq.5b) 6 - D = B
Hint #8
Substitute 1 for F, 6 - D for B (from eq.5b), and 6 for E in eq.6: 1 - D = (6 - D) ÷ 6 Multiply both sides of the above equation by 6: 6 × (1 - D) = 6 × (6 - D) ÷ 6 which becomes 6 - 6×D = 6 - D In the above equation, subtract 6 from both sides, and add D to both sides: 6 - 6×D - 6 + D = 6 - D - 6 + D which simplifies to -5×D = 0 which means D = 0
Hint #9
Substitute 1 for F, and 0 for D in eq.3a: 2×1 + 0 = A which makes 2 = A
Hint #10
Substitute 1 for F, and 0 for D in eq.2b: 8×1 - 0 = C which makes 8 = C
Solution
Substitute 0 for D in eq.5b: 6 - 0 = B which makes 6 = B and makes ABCDEF = 268061