Puzzle for October 1, 2020 ( )
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
Subtract D from both sides of eq.5: E + F – D = A + B + D – D which becomes E + F – D = A + B In eq.2, replace A + B with E + F – D: D + F = E + F – D In the above equation, subtract F from both sides, and add D to both sides: D + F – F + D = E + F – D – F + D which makes 2×D = E
Hint #2
Subtract the left and right sides of eq.6 from the left and right sides of eq.3, respectively: B + F – (F – E) = C + D – (C – D) which is equivalent to B + F – F + E = C + D – C + D which becomes eq.3a) B + E = 2×D
Hint #3
In eq.3a, replace 2×D with E: B + E = E Subtract E from each side of the equation above: B + E – E = E – E which makes B = 0
Hint #4
In eq.5, substitute 2×D for E, and 0 for B: 2×D + F = A + 0 + D Subtract D from both sides of the above equation: 2×D + F – D = A + 0 + D – D which becomes eq.5a) D + F = A
Hint #5
Substitute D + F for A (from eq.5a), and 2×D for E in eq.4: C + F = D + F – C + 2×D which becomes C + F = 3×D + F – C In the above equation, subtract F from both sides, and add C to both sides: C + F – F + C = 3×D + F – C – F + C which makes 2×C = 3×D Divide both sides by 2: 2×C ÷ 2 = 3×D ÷ 2 which makes C = 1½×D
Hint #6
Substitute 0 for B, and 1½×D for C in eq.3: 0 + F = 1½×D + D which makes F = 2½×D
Hint #7
Substitute 2½×D for F, and 0 for B in eq.2: D + 2½×D = A + 0 which makes 3½×D = A
Solution
Substitute 3½×D for A, 0 for B, 1½×D for C, 2×D for E, 2½×D for F in eq.1: 3½×D + 0 + 1½×D + D + 2×D + 2½×D = 21 which simplifies to 10½×D = 21 Divide both sides of the equation above by 10½: 10½×D ÷ 10½ = 21 ÷ 10½ which means D = 2 making A = 3½×D = 3½ × 2 = 7 C = 1½×D = 1½ × 2 = 3 E = 2×D = 2 × 2 = 4 F = 2½×D = 2½ × 2 = 5 and ABCDEF = 703245