Puzzle for October 18, 2021 ( )
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.6, replace B with D + F (from eq.3): D – F = D + F – D + F which becomes D – F = 2×F Add F to both sides of the above equation: D – F + F = 2×F + F which makes D = 3×F
Hint #2
In eq.3, replace D with 3×F: B = 3×F + F which makes B = 4×F
Hint #3
In eq.5, substitute 4×F for B: E + F = 4×F – E In the above equation, subtract F from both sides, and add E to both sides: E + F – F + E = 4×F – E – F + E which makes 2×E = 3×F Divide both sides by 2: 2×E ÷ 2 = 3×F ÷ 2 which makes E = 1½×F
Hint #4
Substitute 3×F for D, and 1½×F for E in eq.2: C = 3×F + 1½×F which makes C = 4½×F
Hint #5
Substitute 4½×F for C in eq.4: 4½×F = A + F Subtract F from each side of the above equation: 4½×F – F = A + F – F which makes 3½×F = A
Solution
Substitute 3½×F for A, 4×F for B, 4½×F for C, 3×F for D, and 1½×F for E in eq.1: 3½×F + 4×F + 4½×F + 3×F + 1½×F + F = 35 which simplifies to 17½×F = 35 Divide both sides of the above equation by 17½: 17½×F ÷ 17½ = 35 ÷ 17½ which means F = 2 making A = 3½×F = 3½ × 2 = 7 B = 4×F = 4 × 2 = 8 C = 4½×F = 4½ × 2 = 9 D = 3×F = 3 × 2 = 6 E = 1½×F = 1½ × 2 = 3 and ABCDEF = 789632