Puzzle for April 29, 2020 ( )
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
Add C and A to both sides of eq.6: D – C + F + C + A = C – A + C + A which becomes eq.6a) D + F + A = 2×C Add C and E to both sides of eq.3: C – E + C + E = D – C + E + C + E which becomes eq.3a) 2×C = D + 2×E
Hint #2
In eq.3a, replace 2×C with D + F + A (from eq.6a): D + F + A = D + 2×E Subtract D from each side of the above equation: D + F + A – D = D + 2×E – D which becomes eq.3b) F + A = 2×E
Hint #3
In eq.3b, replace F with A + E (from eq.5): A + E + A = 2×E which becomes 2×A + E = 2×E Subtract E from both sides of the equation above: 2×A + E – E = 2×E – E which makes 2×A = E
Hint #4
In eq.5, substitute 2×A for E: F = A + 2×A which makes F = 3×A
Hint #5
Substitute 2×A for E in eq.4: A + D – 2×A = B + 2×A which becomes D – A = B + 2×A Add A to both sides of the equation above: D – A + A = B + 2×A + A which becomes eq.4a) D = B + 3×A
Hint #6
Substitute B + 3×A for D (from eq.4a), and 3×A for F in eq.2: B + B + 3×A = 3×A Subtract 3×A from each side of the equation above: B + B + 3×A – 3×A = 3×A – 3×A which makes 2×B = 0 which means B = 0
Hint #7
Substitute 0 for B in eq.4a: D = 0 + 3×A which makes D = 3×A
Hint #8
Substitute 3×A for D and F in eq.6a: 3×A + 3×A + A = 2×C which makes 7×A = 2×C Divide both sides of the equation above by 2: 7×A ÷ 2 = 2×C ÷ 2 which makes 3½×A = C
Solution
Substitute 0 for B, 3½×A for C, 3×A for D and F, and 2×A for E in eq.1: A + 0 + 3½×A + 3×A + 2×A + 3×A = 25 which simplifies to 12½×A = 25 Divide both sides of the equation above by 12½: 12½×A ÷ 12½ = 25 ÷ 12½ which means A = 2 making C = 3½×A = 3½ × 2 = 7 D = F = 3×A = 3 × 2 = 6 E = 2×A = 2 × 2 = 4 and ABCDEF = 207646