Puzzle for January 2, 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
eq.6 may also be written as: B + C = A + F + E From eq.4, substitute C + E for A + F in the above equation: B + C = C + E + E Subtract C from both sides: B + C - C = C + E + E - C which makes B = 2×E
Hint #2
In eq.2, replace B with 2×E: E + F = 2×E + D Subtract E from both sides of the above equation: E + F - E = 2×E + D - E which becomes eq.2a) F = E + D In eq.5, replace F with E + D: A + D = E + D Subtract D from both sides: A + D - D = E + D - D which makes A = E
Hint #3
Substitute E for A, and 2×E for B in eq.6: 2×E + C = E + E + F Subtract 2×E from each side of the equation above: 2×E + C - 2×E = E + E + F - 2×E which makes C = F
Hint #4
Substitute 2×E for B, and F for C in eq.3: 2×E = F + D In the above equation, replace F with E + D (from eq.2a): 2×E = E + D + D Subtract E from each side: 2×E - E = E + D + D - E which simplifies to E = 2×D Divide both sides by 2: E ÷ 2 = 2×D ÷ 2 which means ½×E = D
Hint #5
Substitute ½×E for D, and E for A in eq.5: E + ½×E = F which means 1½×E = F
Solution
Substitute E for A, 2×E for B, 1½×E for C and F, and ½×E for D in eq.1: E + 2×E + 1½×E + ½×E + E + 1½×E = 30 7½×E = 30 Divide both sides of the above equation by 7½: 7½×E ÷ 7½ = 30 ÷ 7½ which makes E = 4 making A = E = 4 B = 2×E = 2 × 4 = 8 C = F = 1½×E = 1½ × 4 = 6 D = ½×E = ½ × 4 = 2 and ABCDEF = 486246