Puzzle for October 17, 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
Add C and E to both sides of eq.3: F – E + C + E = D – C + C + E which becomes F + C = D + E In the above equation, replace D + E with A + F (from eq.2): F + C = A + F Subtract F from both sides: F + C – F = A + F – F which makes C = A
Hint #2
In eq.5, replace C with A, and B + E with D (from eq.4): A + A – E = D which becomes 2×A – E = D Add E to each side of the above equation: 2×A – E + E = D + E which becomes eq.5a) 2×A = D + E
Hint #3
In eq.2, substitute 2×A for D + E (from eq.5a): 2×A = A + F Subtract A from both sides of the equation above: 2×A – A = A + F – A which makes A = F
Hint #4
In eq.6, substitute A for C, and add B to both sides: B + D + B = A – B + A + B which becomes eq.6a) 2×B + D = 2×A
Hint #5
Substitute D + E for 2×A (from eq.5a) in eq.6a: 2×B + D = D + E Subtract D from both sides of the equation above: 2×B + D – D = D + E – D which makes 2×B = E
Hint #6
Substitute 2×B for E in eq.4: B + 2×B = D which makes 3×B = D
Hint #7
Substitute 3×B for D in eq.6a: 2×B + 3×B = 2×A which means 5×B = 2×A Divide both sides of the above equation by 2: 5×B ÷ 2 = 2×A ÷ 2 which makes 2½×B = A and also makes C = F = A = 2½×B
Solution
Substitute 2½×B for A and C and F, 3×B for D, and 2×B for E in eq.1: 2½×B + B + 2½×B + 3×B + 2×B + 2½×B = 27 which simplifies to 13½×B = 27 Divide both sides of the equation above by 13½: 13½×B ÷ 13½ = 27 ÷ 13½ which means B = 2 making A = C = F = 2½×B = 2½ × 2 = 5 D = 3×B = 3 × 2 = 6 E = 2×B = 2 × 2 = 4 and ABCDEF = 525645