Puzzle for October 14, 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.2, replace E with C – D (from eq.4): D + C – D + F = A + C which becomes C + F = A + C Subtract C from both sides of the above equation: C + F – C = A + C – C which makes F = A
Hint #2
In eq.5, replace F with A: A = A + E Subtract A from both sides of the equation above: A – A = A + E – A which means 0 = E
Hint #3
Replace 0 for E, and A for F in eq.2: D + 0 + A = A + C Subtract A from both sides of the above equation: D + 0 + A – A = A + C – A which makes D = C
Hint #4
In eq.3, substitute C for D: C + C = A – C which becomes 2×C = A – C Add C to both sides of the equation above: 2×C + C = A – C + C which means 3×C = A
Hint #5
Substitute 3×C for A and F, and C for D in eq.6: B = (3×C + 3×C – C) ÷ C which becomes B = (5×C) ÷ C which means B = 5
Solution
Substitute 3×C for A and F, 5 for B, C for D, and 0 for E in eq.1: 3×C + 5 + C + C + 0 + 3×C = 29 which simplifies to 8×C + 5 = 29 Subtract 5 from both sides of the above equation: 8×C + 5 – 5 = 29 – 5 which makes 8×C = 24 Divide both sides by 8: 8×C ÷ 8 = 24 ÷ 8 which means C = 3 making A = F = 3×C = 3 × 3 = 9 D = C = 3 and ABCDEF = 953309