Puzzle for October 21, 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
Subtract the left and right sides of eq.4 from the left and right sides of eq.2, respectively: E + F - (D + E) = A + C - (C + F) which becomes E + F - D - E = A + C - C - F which becomes F - D = A - F Add D and F to both sides of the above equation: F - D + D + F = A - F + D + F which becomes eq.2a) 2×F = A + D
Hint #2
Add A and F to both sides of eq.5: F - A + A + F = B + D - F + A + F which becomes eq.5a) 2×F = B + D + A
Hint #3
In eq.5a, replace 2×F with A + D (from eq.2a): A + D = B + D + A Subtract A and D from each side of the equation above: A + D - A - D = B + D + A - A - D which makes 0 = B
Hint #4
Subtract A from both sides of eq.2a: 2×F - A = A + D - A which becomes eq.2b) 2×F - A = D In eq.3, substitute 0 for B, and 2×F - A for D (from eq.2b): A + F = 0 + 2×F - A which becomes A + F = 2×F - A In the above equation, subtract F from both sides, and add A to both sides: A + F - F + A = 2×F - A - F + A which makes 2×A = F
Hint #5
Substitute (2×A) for F in eq.2b: 2×(2×A) - A = D which becomes 4×A - A = D which makes 3×A = D
Hint #6
Substitute 2×A for F in eq.2: E + 2×A = A + C Subtract A from each side of the above equation: E + 2×A - A = A + C - A which becomes eq.2c) E + A = C
Hint #7
Substitute 0 for B, E + A for C (from eq.2c), 3×A for D, and 2×A for F in eq.6: 0 + E + A - 3×A = 3×A - E + 2×A - 0 which becomes E - 2×A = 5×A - E Add 2×A and E to both sides of the above equation: E - 2×A + 2×A + E = 5×A - E + 2×A + E which makes 2×E = 7×A Divide both sides by 2: 2×E ÷ 2 = 7×A ÷ 2 which makes E = 3½×A
Hint #8
Substitute 3½×A for E in eq.2c: 3½×A + A = C which makes 4½×A = C
Solution
Substitute 0 for B, 4½×A for C, 3×A for D, 3½×A for E, and 2×A for F in eq.1: A + 0 + 4½×A + 3×A + 3½×A + 2×A = 28 which simplifies to 14×A = 28 Divide both sides of the above equation by 14: 14×A ÷ 14 = 28 ÷ 14 which means A = 2 making C = 4½×A = 4½ × 2 = 9 D = 3×A = 3 × 2 = 6 E = 3½×A = 3½ × 2 = 7 F = 2×A = 2 × 2 = 4 and ABCDEF = 209674