Puzzle for October 13, 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.1 may be re-written as: B + C + A + D + E + F = 36 In the above equation, replace A + D + E + F with B + C (from eq.5): B + C + B + C = 36 which becomes 2×B + 2×C = 36 Divide both sides by 2: (2×B + 2×C) ÷ 2 = 36 ÷ 2 which makes B + C = 18 Subtract C from both sides: B + C – C = 18 – C which makes eq.1a) B = 18 – C
Hint #2
To make eq.1a true, check several possible one-digit non-negative integers for C and B: If C = 9, then B = 18 – 9 = 9 If C = 8, then B = 18 – 8 = 10 If C < 8, then B > 10 Since B must be a one-digit integer, then B = 9 which means C = 9
Hint #3
In eq.2, substitute 9 for C: 9 + E = A + F Subtract 9 from both sides of the equation above: 9 + E – 9 = A + F – 9 which becomes eq.2a) E = A + F – 9
Hint #4
In eq.5, substitute 9 for B and C, F – A for D (from eq.4), and A + F – 9 for E (from eq.2a): 9 + 9 = A + F – A + A + F – 9 + F which becomes 18 = A + 3×F – 9 In the equation above, add 9 to both sides, and subtract 3×F from both sides: 18 + 9 – 3×F = A + 3×F – 9 + 9 – 3×F which becomes eq.5a) 27 – 3×F = A
Hint #5
Substitute 27 – 3×F for A (from eq.5a) in eq.2a: E = 27 – 3×F + F – 9 which becomes eq.2b) E = 18 – 2×F
Hint #6
Substitute 27 – 3×F for A (from eq.5a) in eq.4: F – D = 27 – 3×F In the above equation, add D to both sides, and subtract (27 – 3×F) from each side: F – D + D – (27 – 3×F) = 27 – 3×F + D – (27 – 3×F) which becomes F – 27 + 3×F = D which becomes eq.4a) 4×F – 27 = D
Solution
Substitute 27 – 3×F for A (from eq.5a), 18 – 2×F for E (from eq.2b), 9 for B, and 4×F – 27 for D (from eq.4a) in eq.3: 27 – 3×F + 18 – 2×F = 9 + 4×F – 27 which becomes 45 – 5×F = 4×F – 18 Add 18 and 5×F to each side of the above equation: 45 – 5×F + 18 + 5×F = 4×F – 18 + 18 + 5×F which simplifies to 63 = 9×F Divide both sides by 9: 63 ÷ 9 = 9×F ÷ 9 which means 7 = F making A = 27 – 3×F = 27 – 3×7 = 27 – 21 = 6 (from eq.5a) D = 4×F – 27 = 4×7 – 27 = 28 – 27 = 1 (from eq.4a) E = 18 – 2×F = 18 – 2×7 = 18 – 14 = 4 (from eq.2b) and ABCDEF = 699147