Puzzle for September 10, 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
eq.6 may be written as: D + F - C = A - (D - E) In the above equation, replace D - E with B - F (from eq.3): D + F - C = A - (B - F) which becomes eq.6a) D + F - C = A - B + F
Hint #2
Subtract F from both sides of eq.6a: D + F - C - F = A - B + F - F which becomes D - C = A - B Add C and B to both sides of the equation above: D - C + C + B = A - B + C + B which becomes D + B = A + C which may be written as eq.6b) B + D = A + C
Hint #3
In eq.6b, replace B + D with A (from eq.2): A = A + C Subtract A from each side of the above equation: A - A = A + C - A which makes 0 = C
Hint #4
In eq.4, substitute 0 for C: E - D = A + 0 - E which becomes E - D = A - E Add E to both sides of the equation above: E - D + E = A - E + E which becomes eq.4a) 2×E - D = A
Hint #5
Substitute 2×E - D for A (from eq.4a) in eq.2: 2×E - D = B + D Subtract D from each side of the equation above: 2×E - D - D = B + D - D which becomes eq.2a) 2×E - 2×D = B
Hint #6
Substitute 2×E - 2×D for B (from eq.2a) into eq.3: 2×E - 2×D - F = D - E Add F and E to both sides of the above equation: 2×E - 2×D - F + F + E = D - E + F + E which becomes 3×E - 2×D = D + F Subtract D from both sides: 3×E - 2×D - D = D + F - D which becomes eq.3a) 3×E - 3×D = F
Hint #7
Substitute 2×E - D for A (from eq.4a), 2×E - 2×D for B (from eq.2a), and 3×E - 3×D for F (from eq.3a) in eq.5: D + E = 2×E - D + 2×E - 2×D + 3×E - 3×D which becomes D + E = 7×E - 6×D In the equation above, subtract E from both sides, and add 6×D to both sides: D + E - E + 6×D = 7×E - 6×D - E + 6×D which becomes eq.5a) 7×D = 6×E
Hint #8
Multiply both sides of eq.2a by 3: 3 × (2×E - 2×D) = 3 × B which becomes 6×E - 6×D = 3×B Substitute 7×D for 6×E (from eq.5a) in the above equation: 7×D - 6×D = 3×B which makes D = 3×B
Hint #9
Substitute (3×B) for D in eq.5a: 7×(3×B) = 6×E which makes 21×B = 6×E Divide both sides of the above equation by 6: 21×B ÷ 6 = 6×E ÷ 6 which makes 3½×B = E
Hint #10
Substitute (3½×B) for E, and (3×B) for D in eq.3a: 3×(3½×B) - 3×(3×B) = F which becomes 10½×B - 9×B = F which makes 1½×B = F
Hint #11
Substitute (3½×B) for E, and 3×B for D in eq.4a: 2×(3½×B) - 3×B = A which becomes 7×B - 3×B = A which makes 4×B = A
Solution
Substitute 4×B for A, 0 for C, 3×B for D, 3½×B for E, and 1½×B for F in eq.1: 4×B + B + 0 + 3×B + 3½×B + 1½×B = 26 which simplifies to 13×B = 26 Divide both sides of the above equation by 13: 13×B ÷ 13 = 26 ÷ 13 which means B = 2 making A = 4×B = 4 × 2 = 8 D = 3×B = 3 × 2 = 6 E = 3½×B = 3½ × 2 = 7 F = 1½×B = 1½ × 2 = 3 and ABCDEF = 820673