Puzzle for March 28, 2019 ( )
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 D + E with B (from eq.5): B + C = B + F Subtract B from each side of the above equation: B + C - B = B + F - B which makes C = F
Hint #2
eq.2 may be written as: A + B = C + E + F Subtract the left and right sides of eq.3 from the left and right sides of the equation above, respectively: A + B - (B + C) = C + E + F - (D + E + F) which is equivalent to A + B - B - C = C + E + F - D - E - F which simplifies to A - C = C - D Add C to both sides: A - C + C = C - D + C which becomes eq.3a) A = 2×C - D
Hint #3
In eq.6, replace B with D + E (from eq.5): A = D + E + D which becomes A = 2×D + E In eq.3a, substitute 2×D + E for A: 2×D + E = 2×C - D Subtract 2×D from both sides of the above equation: 2×D + E - 2×D = 2×C - D - 2×D which becomes eq.3b) E = 2×C - 3×D
Hint #4
Substitute 2×C - D for A (from eq.3a), and 2×C - 3×D for E (from eq.3b) in eq.4: 2×C - D + 2×C - 3×D = C + D which becomes 4×C - 4×D = C + D Add (4×D - C) to each side of the above equation: 4×C - 4×D + (4×D - C) = C + D + (4×D - C) which simplifies to 3×C = 5×D Divide both sides by 3: 3×C ÷ 3 = 5×D ÷ 3 which means C = 1⅔×D which also means F = C = 1⅔×D
Hint #5
Substitute (1⅔×D) for C in eq.3a: A = 2×(1⅔×D) - D which is equivalent to A = 3⅓×D - D which means A = 2⅓×D
Hint #6
Substitute (1⅔×D) for C in eq.3b: E = 2×(1⅔×D) - 3×D which is equivalent to E = 3⅓×D - 3×D which makes E = ⅓×D
Hint #7
In eq.5, substitute ⅓×D for E: D + ⅓×D = B which makes 1⅓×D = B
Solution
Substitute 2⅓×D for A, 1⅓×D for B, 1⅔×D for C and F, and ⅓×D for E in eq.1: 2⅓×D + 1⅓×D + 1⅔×D + D + ⅓×D + 1⅔×D = 25 which becomes 8⅓×D = 25 Divide both sides of the equation above by 8⅓: 8⅓×D ÷ 8⅓ = 25 ÷ 8⅓ which means D = 3 making A = 2⅓×D = 2⅓ × 3 = 7 B = 1⅓×D = 1⅓ × 3 = 4 C = F = 1⅔×D = 1⅔ × 3 = 5 E = ⅓×D = ⅓ × 3 = 3 and ABCDEF = 745315