Puzzle for February 28, 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.
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Hint #1
In eq.4, replace F with A + B (from eq.2): B + C = D + A + B - A which becomes B + C = D + B Subtract B from each side of the above equation: B + C - B = D + B - B which makes C = D
Hint #2
In eq.5, replace D with C: C - A = A - C - C which becomes C - A = A - 2×C Add A and 2×C to both sides of the above equation: C - A + A + 2×C = A - 2×C + A + 2×C which makes 3×C = 2×A Divide both sides by 2: 3×C ÷ 2 = 2×A ÷ 2 which makes 1½×C = A
Hint #3
In eq.3, substitute 1½×C for A: 1½×C - B = B - E Add B and E to both sides of the above equation: 1½×C - B + B + E = B - E + B + E which becomes 1½×C + E = 2×B Subtract 1½×C from each side: 1½×C + E - 1½×C = 2×B - 1½×C which becomes eq.3a) E = 2×B - 1½×C
Hint #4
Substitute C for D, 1½×C for A, and 2×B - 1½×C for E (from eq.3a) in eq.6: B + C + C = 1½×C + 2×B - 1½×C - C which becomes B + 2×C = 2×B - C In the above equation, subtract B from both sides, and add C to both sides: B + 2×C - B + C = 2×B - C - B + C which makes 3×C = B
Hint #5
Substitute (3×C) for B in eq.3a: E = 2×(3×C) - 1½×C which becomes E = 6×C - 1½×C which makes E = 4½×C
Hint #6
Substitute 1½×C for A, and 3×C for B in eq.2: F = 1½×C + 3×C which makes F = 4½×C
Solution
Substitute 1½×C for A, 3×C for B, C for D, and 4½×C for E and F in eq.1: 1½×C + 3×C + C + C + 4½×C + 4½×C = 31 which simplifies to 15½×C = 31 Divide both sides of the above equation by 15½: 15½×C ÷ 15½ = 31 ÷ 15½ which means C = 2 making A = 1½×C = 1½ × 2 = 3 B = 3×C = 3 × 2 = 6 D = C = 2 E = F = 4½×C = 4½ × 2 = 9 and ABCDEF = 362299