Puzzle for September 11, 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.3, replace C with E + F (from eq.2): E + F + E = A which becomes eq.3a) 2×E + F = A
Hint #2
In eq.4, replace A with 2×E + F (from eq.3a): D + F = 2×E + F Subtract F from each side of the equation above: D + F – F = 2×E + F – F which makes eq.4a) D = 2×E
Hint #3
In eq.5, substitute 2×E for D (from eq.4a): B + E + F = 2×E Subtract E from both sides of the equation above: B + E + F – E = 2×E – E which becomes eq.5a) B + F = E
Hint #4
Substitute B + F for E (from eq.5a) in eq.2: B + F + F = C which becomes eq.2a) B + 2×F = C
Hint #5
Substitute (B + F) for E (from eq.5a) in eq.3a: 2×(B + F) + F = A which is equivalent to 2×B + 2×F + F = A which becomes eq.3b) 2×B + 3×F = A
Hint #6
Substitute (B + F) for E (from eq.5a) in eq.4a: D = 2×(B + F) which is equivalent to eq.4b) D = 2×B + 2×F
Hint #7
Substitute (2×B + 3×F) for A (from eq.3b), B + 2×F for C (from eq.2a), 2×B + 2×F for D (from eq.4b), and B + F for E (from eq.5a) into eq.6: (2×B + 3×F) + F = B + B + 2×F + 2×B + 2×F + B + F – (2×B + 3×F) – F which becomes 2×B + 4×F = 5×B + 4×F – 2×B – 3×F which becomes 2×B + 4×F = 3×B + F Subtract 2×B and F from each side of the equation above: 2×B + 4×F – 2×B – F = 3×B + F – 2×B – F which makes 3×F = B
Hint #8
Substitute (3×F) for B in eq.3b: 2×(3×F) + 3×F = A which becomes 6×F + 3×F = A which makes 9×F = A
Hint #9
Substitute 3×F for B in eq.2a: 3×F + 2×F = C which makes 5×F = C
Hint #10
Substitute (3×F) for B in eq.4b: D = 2×(3×F) + 2×F which becomes D = 6×F + 2×F which makes D = 8×F
Hint #11
Substitute 3×F for B in eq.5a: 3×F + F = E which makes 4×F = E
Solution
Substitute 9×F for A, 3×F for B, 5×F for C, 8×F for D, and 4×F for E in eq.1: 9×F + 3×F + 5×F + 8×F + 4×F + F = 30 which simplifies to 30×F = 30 Divide both sides of the above equation by 30: 30×F ÷ 30 = 30 ÷ 30 which means F = 1 making A = 9×F = 9 × 1 = 9 B = 3×F = 3 × 1 = 3 C = 5×F = 5 × 1 = 5 D = 8×F = 8 × 1 = 8 E = 4×F = 4 × 1 = 4 and ABCDEF = 935841