Puzzle for June 13, 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
In eq.3, replace B - A with A - E (from eq.4): F - B = A - E Add B and E to both sides of the above equation: F - B + B + E = A - E + B + E which becomes eq.3a) F + E = A + B
Hint #2
In eq.5, replace A + B with F + E (from eq.3a): E + F = F + E - E which becomes E + F = F Subtract F from each side of the equation above: E + F - F = F - F which makes E = 0
Hint #3
In eq.4, substitute 0 for E: B - A = A - 0 which becomes B - A = A Add A to both sides of the equation above: B - A + A = A + A which makes B = 2×A
Hint #4
Substitute 2×A for B in eq.3: F - 2×A = 2×A - A which becomes F - 2×A = A Add 2×A to both sides of the above equation: F - 2×A + 2×A = A + 2×A which makes F = 3×A
Hint #5
Substitute 3×A for F, and 2×A for B in eq.2: D - 3×A = 3×A - 2×A which becomes D - 3×A = A Add 3×A to both sides of the above equation: D - 3×A + 3×A = A + 3×A which makes D = 4×A
Hint #6
Substitute 0 for E, 2×A for B, 4×A for D, and 3×A for F in eq.6: C + 0 = 2×A - C + 4×A - 0 + 3×A which becomes C = 9×A - C Add C to both sides of the equation above: C + C = 9×A - C + C which makes 2×C = 9×A Divide both sides by 2: 2×C ÷ 2 = 9×A ÷ 2 which makes C = 4½×A
Solution
Substitute 2×A for B, 4½×A for C, 4×A for D, 0 for E, and 3×A for F in eq.1: A + 2×A + 4½×A + 4×A + 0 + 3×A = 29 which simplifies to 14½×A = 29 Divide both sides of the above equation by 14½: 14½×A ÷ 14½ = 29 ÷ 14½ which means A = 2 making B = 2×A = 2 × 2 = 4 C = 4½×A = 4½ × 2 = 9 D = 4×A = 4 × 2 = 8 F = 3×A = 3 × 2 = 6 and ABCDEF = 249806