Puzzle for August 9, 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.4, replace E with A + B (from eq.3): A + C = B + A + B which becomes A + C = A + 2×B Subtract A from both sides of the above equation: A + C – A = A + 2×B – A which makes C = 2×B
Hint #2
In eq.2, replace A + B with E (from eq.3): E + C + D = E + F Subtract E from both sides of the equation above: E + C + D – E = E + F – E which becomes eq.2a) C + D = F
Hint #3
In eq.5, substitute 2×B for C, and A + B for E (from eq.3): A + B + 2×B + A + B = D which becomes eq.5a) 2×A + 4×B = D
Hint #4
Substitute 2×B for C, and 2×A + 4×B for D (from eq.5a) in eq.2a: 2×B + 2×A + 4×B = F which becomes eq.2b) 2×A + 6×B = F
Hint #5
Substitute 2×B for C, 2×A + 4×B for D (from eq.5a), A + B for E (from eq.3), and 2×A + 6×B for F (from eq.2b) in eq.1: A + B + 2×B + 2×A + 4×B + A + B + 2×A + 6×B = 20 which simplifies to 6×A + 14×B = 20 Subtract 14×B from each side of the above equation: 6×A + 14×B – 14×B = 20 – 14×B which becomes 6×A = 20 – 14×B Divide both sides by 6: 6×A ÷ 6 = (20 – 14×B) ÷ 6 which becomes eq.1a) A = (20 – 14×B) ÷ 6
Hint #6
To make eq.1a true, check several possible values for B and A: If B = 0, then A = (20 – 14×0) ÷ 6 = (20 – 0) ÷ 6 = 20 ÷ 6 = 3.33333333 If B = 1, then A = (20 – 14×1) ÷ 6 = (20 – 14) ÷ 6 = 6 ÷ 6 = 1 If B = 2, then A = (20 – 14×2) ÷ 6 = (20 – 28) ÷ 6 = –8 ÷ 6 = –1.33333333 If B > 2, then A < –1.33333333 Since A must be a non-negative integer, then A ≠ 3.33333333 or –1.33333333 which makes A = 1 and also makes B = 1
Solution
Since A = 1 and B = 1, then C = 2×B = 2×1 = 2 D = 2×A + 4×B = 2×1 + 4×1 = 2 + 4 = 6 (from eq.5a) E = A + B = 1 + 1 = 2 (from eq.3) F = 2×A + 6×B = 2×1 + 6×1 = 2 + 6 = 8 (from eq.2b) and ABCDEF = 112628