Puzzle for August 24, 2019 ( )
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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.
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Hint #1
Add E to both sides of eq.4: D – E + E = A + E which makes eq.4a) D = A + E
Hint #2
eq.1 may be written as: A + E + B + C + F + D = 25 In the above equation, replace A + E with D (from eq.4a), and replace B + C + F with D – C (from eq.5): D + D – C + D = 25 which becomes 3×D – C = 25 Add C to both sides, and subtract 25 from both sides: 3×D – C + C – 25 = 25 + C – 25 which becomes eq.1a) 3×D – 25 = C
Hint #3
To make eq.1a true, check several possible values for D and C: If D = 9, then C = 3×9 – 25 = 27 – 25 = 2 If D = 8, then C = 3×8 – 25 = 24 – 25 = –1 If D < 8, then C < –1 Since C must be a non-negative integer, then C = 2 which makes D = 9
Hint #4
In eq.4, substitute 9 for D: eq.4b) 9 – E = A
Hint #5
Substitute 9 – E for A (from eq.4b) in eq.2: E + F = 9 – E Subtract E from both sides: E + F – E = 9 – E – E which makes eq.2a) F = 9 – 2×E
Hint #6
Substitute 2 for C, and 9 – 2×E for F (from eq.2a) in eq.3: 2 + E = B + 9 – 2×E In the above equation, add 2×E to both sides, and subtract 9 from both sides: 2 + E + 2×E – 9 = B + 9 – 2×E + 2×E – 9 which simplifies to eq.3a) 3×E – 7 = B
Solution
Substitute 9 – E for A (from eq.4b), 3×E – 7 for B (from eq.3a), 2 for C, 9 for D, and 9 – 2×E for F (from eq.2a) in eq.1: 9 – E + 3×E – 7 + 2 + 9 + E + 9 – 2×E = 25 which simplifies to E + 22 = 25 Subtract 22 from both sides of the above equation: E + 22 – 22 = 25 – 22 which means E = 3 making A = 9 – E = 9 – 3 = 6 (from eq.4b) B = 3×E – 7 = 3×3 – 7 = 9 – 7 = 2 (from eq.3a) F = 9 – 2×E = 9 – 2×3 = 9 – 6 = 3 (from eq.2a) and ABCDEF = 622933