Puzzle for November 9, 2022 ( )
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, substitute (C + D) for E (from eq.3): D + (C + D) = B + C - (C + D) which becomes 2×D + C = B + C - C - D which becomes 2×D + C = B - D Add D to both sides of the above equation: 2×D + C + D = B - D + D which becomes eq.5a) 3×D + C = B
Hint #2
In eq.2, replace B with 3×D + C (from eq.5a), and E with C + D (from eq.3): 3×D + C = C + D + F Subtract C and D from each side of the equation above: 3×D + C - C - D = C + D + F - C - D which simplifies to 2×D = F
Hint #3
eq.6 may be written as: A = (B + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (B + E + F) ÷ 3 which becomes eq.6a) 3×A = B + E + F
Hint #4
In eq.6a, replace E + F with B (from eq.2): 3×A = B + B which makes 3×A = 2×B Divide both sides of the above equation by 2: 3×A ÷ 2 = 2×B ÷ 2 which makes eq.6b) 1½×A = B
Hint #5
In eq.4, substitute 1½×A for B, and 2×D for F: 1½×A - D = A + 2×D In the above equation, add D to both sides, and subtract A from both sides: 1½×A - D + D - A = A + 2×D + D - A which makes ½×A = 3×D Multiply both sides by 2: 2 × ½×A = 2 × 3×D which makes A = 6×D
Hint #6
Substitute (6×D) for A in eq.6b: 1½×(6×D) = B which makes 9×D = B
Hint #7
Substitute 9×D for B in eq.5a: 3×D + C = 9×D Subtract 3×D from each side of the above equation: 3×D + C - 3×D = 9×D - 3×D which makes C = 6×D
Hint #8
Substitute 6×D for C in eq.3: E = 6×D + D which makes E = 7×D
Solution
Substitute 6×D for A and C, 9×D for B, 7×D for E, and 2×D for F in eq.1: 6×D + 9×D + 6×D + D + 7×D + 2×D = 31 which simplifies to 31×D = 31 Divide both sides of the above equation by 31: 31×D ÷ 31 = 31 ÷ 31 which means D = 1 making A = C = 6×D = 6 × 1 = 6 B = 9×D = 9 × 1 = 9 E = 7×D = 7 × 1 = 7 F = 2×D = 2 × 1 = 2 and ABCDEF = 696172