Puzzle for August 10, 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 (from eq.3): E – A = A + E – E which becomes E – A = A Add A to both sides of the equation above: E – A + A = A + A which makes E = 2×A
Hint #2
In eq.3, replace E with 2×A: A + 2×A = B which makes 3×A = B
Hint #3
In eq.5, substitute 3×A for B: 3×A – F = A + F In the equation above, add F to both sides, and subtract A from both sides: 3×A – F + F – A = A + F + F – A which makes 2×A = 2×F Divide both sides by 2: 2×A ÷ 2 = 2×F ÷ 2 which makes A = F
Hint #4
Substitute 3×A for B, and 2×A for E in eq.6: C – 3×A – 2×A = A + 3×A – C which becomes C – 5×A = 4×A – C Add 5×A and C to both sides of the above equation: C – 5×A + 5×A + C = 4×A – C + 5×A + C which becomes 2×C = 9×A Divide both sides by 2: 2×C ÷ 2 = 9×A ÷ 2 which makes C = 4½×A
Hint #5
Substitute A for F, and 4½×A for C in eq.2: D + A = A + 4½×A which becomes D + A = 5½×A Subtract A from each side of the equation above: D + A – A = 5½×A – A which makes D = 4½×A
Solution
Substitute 3×A for B, and 4½×A for C and D, 2×A for E, and A for F in eq.1: A + 3×A + 4½×A + 4½×A + 2×A + A = 32 which simplifies to 16×A = 32 Divide both sides of the equation above by 16: 16×A ÷ 16 = 32 ÷ 16 which means A = 2 making B = 3×A = 3 × 2 = 6 C = D = 4½×A = 4½ × 2 = 9 E = 2×A = 2 × 2 = 4 F = A = 2 and ABCDEF = 269942