Puzzle for March 30, 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
In eq.4, replace B with A + E + F (from eq.6): A + A + E + F = D + E + F which becomes 2×A + E + F = D + E + F Subtract E and F from each side of the above equation: 2×A + E + F – E – F = D + E + F – E – F which simplifies to 2×A = D
Hint #2
In eq.2, replace D with 2×A: 2×A = A + E Subtract A from each side of the equation above: 2×A – A = A + E – A which makes A = E
Hint #3
In eq.3, substitute A for E, and 2×A for D: A + F = 2×A Subtract A from both sides of the above equation: A + F – A = 2×A – A which makes F = A
Hint #4
In eq.6, substitute A for both E and F: B = A + A + A which means B = 3×A
Hint #5
Substitute 2×A for D, and 3×A for B in eq.5: C + 2×A = 3×A – C In the above equation, add C to both sides, and subtract 2×A from each side: C + 2×A + C – 2×A = 3×A – C + C – 2×A which makes 2×C = A Divide both sides of the above equation by 2: 2×C ÷ 2 = A ÷ 2 which means C = ½×A
Solution
Substitute 3×A for B, ½×A for C, 2×A for D, and A for E and F in eq.1: A + 3×A + ½×A + 2×A + A + A = 17 which simplifies to 8½×A = 17 Divide both sides of the equation above by 8½: 8½×A ÷ 8½ = 17 ÷ 8½ which means A = 2 making B = 3×A = 3 × 2 = 6 C = ½×A = ½ × 2 = 1 D = 2×A = 2 × 2 = 4 E = F = A = 2 and ABCDEF = 261422