Puzzle for September 10, 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 E with C + D (from eq.3): B + C = A – D + C + D which becomes B + C = A + C Subtract C from both sides of the above equation: B + C – C = A + C – C which makes B = A
Hint #2
In eq.2, replace B with A: A + D = A + C which becomes Subtract A from both sides of the equation above: A + D – A = A + C – A which makes D = C
Hint #3
In eq.3, substitute C for D: C + C = E which makes eq.3a) 2×C = E
Hint #4
Substitute C for D, 2×C for E, A for B, and A + C for F (from eq.4) in eq.5: C + 2×C = A + A + C + A + C which becomes 3×C = 3×A + 2×C Subtract 2×C from both sides: 3×C – 2×C = 3×A + 2×C – 2×C which makes C = 3×A and also makes D = C = 3×A
Hint #5
Substitute (3×A) for C in eq.3a: 2×(3×A) = E which means 6×A = E
Hint #6
Substitute 3×A for C in eq.4: F = A + 3×A which makes F = 4×A
Solution
Substitute A for B, 3×A for C for D, 6×A for E, and 4×A for F in eq.1: A + A + 3×A + 3×A + 6×A + 4×A = 18 which simplifies to 18×A = 18 Divide both sides of the equation above by 18: 18×A ÷ 18 = 18 ÷ 18 which means A = 1 making B = A = 1 C = D = 3×A = 3×1 = 3 E = 6×A = 6×1 = 6 F = 4×A = 4×1 = 4 and ABCDEF = 113364