Puzzle for December 27, 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.1, replace A + B + C + D with E + F (from eq.2): E + F + E + F = 24 which may be written as 2×(E + F) = 24 Divide both sides of the above equation by 2: 2×(E + F) ÷ 2 = 24 ÷ 2 which becomes eq.1a) E + F = 12 Subtract F from both sides of eq.1a: E + F – F = 12 – F which makes eq.1b) E = 12 – F
Hint #2
In eq.4, replace E with 12 – F (from eq.1b): C + 12 – F = F In the equation above, add F to each side, and subtract 12 from each side: C + 12 – F + F – 12 = F + F – 12 which makes eq.4a) C = 2×F – 12
Hint #3
Subtract C and D from both sides of eq.5: B + D – C – D = C + F – C – D which becomes B – C = F – D In eq.3, substitute F – D for B – C: D – E = F – D Add D and E to both sides of the above equation: D – E + D + E = F – D + D + E which becomes eq.5a) 2×D = E + F
Hint #4
Substitute 12 for E + F (from eq.1a) in eq.5a: 2×D = 12 Divide both sides of the above equation by 2: 2×D ÷ 2 = 12 ÷ 2 which makes D = 6
Hint #5
Substitute 6 for D in eq.6: F = A + 6 Subtract 6 from both sides of the above equation: F – 6 = A + 6 – 6 which makes eq.6a) F – 6 = A
Hint #6
Substitute 6 for D, (12 – F) for E (from eq.1b), and (2×F – 12) for C (from eq.4a) in eq.3: 6 – (12 – F) = B – (2×F – 12) which becomes 6 – 12 + F = B – 2×F + 12 which becomes eq.3a) 3×F – 18 = B
Solution
Substitute F – 6 for A (from eq.6a), 3×F – 18 for B (from eq.3a), 2×F – 12 for C (from eq.4a), 6 for D, and 12 – F for E (from eq.1b) in eq.1: F – 6 + 3×F – 18 + 2×F – 12 + 6 + 12 – F + F = 24 which simplifies to 6×F – 18 = 24 Add 18 to each side of the equation above: 6×F – 18 + 18 = 24 + 18 which becomes 6×F = 42 Divide both sides by 6: 6×F ÷ 6 = 42 ÷ 6 which means F = 7 making A = F – 6 = 7 – 6 = 1 (from eq.6a) B = 3×F – 18 = 3×7 – 18 = 21 – 18 = 3 (from eq.3a) C = 2×F – 12 = 2×7 – 12 = 14 – 12 = 2 (from eq.4a) E = 12 – F = 12 – 7 = 5 (from eq.1b) and ABCDEF = 132657