Puzzle for September 20, 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.6, replace A + B + D with E (from eq.3): E + F = C + E Subtract E from each side of the above equation: E + F – E = C + E – E which makes F = C
Hint #2
In eq.2, replace F with C: C + D = A + C Subtract C from both sides of the above equation: C + D – C = A + C – C which makes D = A
Hint #3
eq.6 may be written as: B + A + D + F = C + E In the above equation, substitute B for A + D + F (from eq.5): B + B = C + E which becomes eq.6a) 2×B = C + E
Hint #4
eq.1 may be written as: A + B + D + F + C + E = 28 Substitute C + E for A + B + D + F (from eq.6) in the equation above: C + E + C + E = 28 which may be written as eq.1a) 2×(C + E) = 28
Hint #5
Substitute 2×B for C + E (from eq.6a) in eq.1a: 2×(2×B) = 28 which becomes 4×B = 28 Divide both sides of the above equation by 4: 4×B ÷ 4 = 28 ÷ 4 which means B = 7
Hint #6
Substitute 7 for B, and A for D in eq.3: E = A + 7 + A which makes eq.3a) E = 2×A + 7
Hint #7
Substitute A for D, 7 for B, and C for F in eq.5: 7 = A + A + C which becomes eq.5a) 7 = 2×A + C Substitute 2×A + C for 7 (from eq.5a) in eq.3a: E = 2×A + 2×A + C which means eq.3b) E = 4×A + C
Hint #8
Substitute A for D, 4×A + C for E (from eq.3b), and C for F in eq.4: A + 4×A + C = C + C which becomes 5×A + C = 2×C Subtract C from both sides of the above equation: 5×A + C – C = 2×C – C which makes 5×A = C and also makes F = C = 5×A
Solution
Substitute 5×A for C in eq.5a: 7 = 2×A + 5×A which means 7 = 7×A Divide both sides of the above equation by 7: 7 ÷ 7 = 7×A ÷ 7 which makes 1 = A making C = F = 5×A = 5 × 1 = 5 D = A = 1 E = 4×A + C = 4×1 + 5 = 9 (from eq.3b) and ABCDEF = 175195