Puzzle for December 5, 2018 ( )
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
From eq.5, substitute A + C for D in eq.2: A + C = A + F Subtract A from each side of the equation above: A + C - A = A + F - A which makes C = F
Hint #2
From eq.4, substitute A + E for B + C in eq.3: A + D = A + E + F Subtract A from each side: A + D - A = A + E + F - A which becomes D = E + F From eq.2, substitute A + F for D in the equation above: A + F = E + F Subtract F from each side: A + F - F = E + F - F which makes A = E
Hint #3
Add B to each side of eq.6: B - C + B = D - B + B which makes eq.6a) 2×B - C = D Substitute C for F, and 2×B - C for D (from eq.6a) in eq.3: A + 2×B - C = B + C + C Add (C - 2×B) to both sides of the above equation: A + 2×B - C + (C - 2×B) = B + C + C + (C - 2×B) which becomes eq.3a) A = 3×C - B
Hint #4
In eq.2, substitute 3×C - B for A (from eq.3a), 2×B - C for D (from eq.6a), and C for F: 2×B - C = 3×C - B + C Add B + C to both sides of the above equation: 2×B - C + B + C = 3×C - B + C + B + C which simplifies to 3×B = 5×C Divide each side by 3: 3×B ÷ 3 = 5×C ÷ 3 which makes B = 1⅔×C
Hint #5
Substitute 1⅔×C for B in eq.3a: A = 3×C - 1⅔×C which makes A = 1⅓×C
Hint #6
Substitute (1⅔×C) for B in eq.6a: 2×(1⅔×C) - C = D which becomes 3⅓×C - C = D which makes 2⅓×C = D
Solution
Substitute 1⅓×C for A and E, 1⅔×C for B, 2⅓×C for D, and C for F in eq.1: 1⅓×C + 1⅔×C + C + 2⅓×C + 1⅓×C + C = 26 which becomes 8⅔×A = 26 Divide each side by 8⅔: 8⅔×A ÷ 8⅔ = 26 ÷ 8⅔ which makes A = 3 making A = E = 1⅓×C = 1⅓ × 3 = 4 B = 1⅔×C = 1⅔ × 3 = 5 D = 2⅓×C = 2⅓ × 3 = 7 F = C = 3 and ABCDEF = 453743