Puzzle for November 8, 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.4, replace C with B + E (from eq.6): D + E = B + B + E which becomes D + E = 2×B + E Subtract E from each side of the above equation: D + E - E = 2×B + E - E which makes eq.4a) D = 2×B
Hint #2
In eq.2, replace D with 2×B: eq.2a) C = A + 2×B
Hint #3
In eq.5, substitute A + 2×B for C (from eq.2a): B + F = A + A + 2×B which becomes B + F = 2×A + 2×B Subtract B from both sides of the equation above: B + F - B = 2×A + 2×B - B which becomes eq.5a) F = 2×A + B
Hint #4
Substitute 2×A + B for F (from eq.5a) in eq.3: B + D = A + 2×A + B which becomes B + D = 3×A + B Subtract B from both sides of the above equation: B + D - B = 3×A + B - B which makes D = 3×A
Hint #5
Substitute 3×A for D in eq.2: C = A + 3×A which makes C = 4×A
Hint #6
Substitute 3×A for D in eq.4a: 3×A = 2×B Divide both sides of the above equation by 2: 3×A ÷ 2 = 2×B ÷ 2 which makes 1½×A = B
Hint #7
Substitute 1½×A for B in eq.5a: F = 2×A + 1½×A which makes F = 3½×A
Hint #8
Substitute 4×A for C, and 1½×A for B in eq.6: 4×A = 1½×A + E Subtract 1½×A from each side of the equation above: 4×A - 1½×A = 1½×A + E - 1½×A which makes 2½×A = E
Solution
Substitute 1½×A for B, 4×A for C, 3×A for D, 2½×A for E, and 3½×A for F in eq.1: A + 1½×A + 4×A + 3×A + 2½×A + 3½×A = 31 which simplifies to 15½×A = 31 Divide both sides of the above equation by 15½: 15½×A ÷ 15½ = 31 ÷ 15½ which means A = 2 making B = 1½×A = 1½ × 2 = 3 C = 4×A = 4 × 2 = 8 D = 3×A = 3 × 2 = 6 E = 2½×A = 2½ × 2 = 5 F = 3½×A = 3½ × 2 = 7 and ABCDEF = 238657