Puzzle for October 22, 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 be written as: E = D + C + F In the equation above, replace C + F with D (from eq.4): E = D + D which makes eq.6a) E = 2×D
Hint #2
In eq.2, replace E with 2×D: D + 2×D = A which makes eq.2a) 3×D = A
Hint #3
In eq.3, substitute C + F for D (from eq.4): B – F = C + F + F Add F to both sides of the above equation: B – F + F = C + F + F + F which becomes eq.3a) B = C + 3×F
Hint #4
Substitute 3×D for A, and C + 3×F for B (from eq.3a) in eq.5: 3×D + C = C + 3×F + D Subtract C and D from both sides of the above equation: 3×D + C – C – D = C + 3×F + D – C – D which makes 2×D = 3×F Divide both sides by 2: 2×D ÷ 2 = 3×F ÷ 2 which means D = 1½×F
Hint #5
Substitute 1½×F for D in eq.6a: E = 2×(1½×F) which makes E = 3×F
Hint #6
Substitute 1½×F for D in eq.2a: 3×(1½×F) = A which makes 4½×F = A
Hint #7
Substitute 1½×F for D in eq.4: C + F = 1½×F Subtract F from each side of the above equation: C + F – F = 1½×F – F C = ½×F
Hint #8
Substitute ½×F for C in eq.3a: B = ½×F + 3×F which makes B = 3½×F
Solution
Substitute 4½×F for A, 3½×F for B, ½×F for C, 1½×F for D, and 3×F for E in eq.1: 4½×F + 3½×F + ½×F + 1½×F + 3×F + F = 28 which simplifies to 14×F = 28 Divide both sides of the above equation by 14: 14×F ÷ 14 = 28 ÷ 14 which means F = 2 making A = 4½×F = 4½ × 2 = 9 B = 3½×F = 3½ × 2 = 7 C = ½×F = ½ × 2 = 1 D = 1½×F = 1½ × 2 = 3 E = 3×F = 3 × 2 = 6 and ABCDEF = 971362