Puzzle for December 5, 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 B with A + E (from eq.6): D + E = A + A + E which becomes D + E = 2×A + E Subtract E from both sides of the above equation: D + E – E = 2×A + E – E which makes D = 2×A
Hint #2
In eq.3, replace D with 2×A: eq.3a) E + F = 2×A
Hint #3
In eq.4, replace D with 2×A, and E + F with 2×A (from eq.3a): B + C = 2×A + 2×A which makes eq.4a) B + C = 4×A
Hint #4
In eq.1, substitute 4×A for B + C (from eq.4a), and 2×A for D and for E + F (from eq.3a): A + 4×A + 2×A + 2×A = 27 which makes 9×A = 27 Divide both sides of the above equation by 9: 9×A ÷ 9 = 27 ÷ 9 which means A = 3 making D = 2×3 = 6
Hint #5
Substitute 6 for D, and 3 for A in eq.2: 6 + E = 3 + B Subtract 3 from each side of the above equation: 6 + E – 3 = 3 + B – 3 which makes eq.2a) 3 + E = B
Hint #6
Substitute 3 + E for B (from eq.2a), and 3 for A in eq.4a: 3 + E + C = 4×3 which is equivalent to 3 + E + C = 12 Subtract 3 and E from each side of the equation above: 3 + E + C – 3 – E = 12 – 3 – E which makes eq.4b) C = 9 – E
Hint #7
Substitute 3 for A in eq.3a: E + F = 2×3 which is equivalent to E + F = 6 Subtract E from both sides of the above equation: E + F – E = 6 – E which makes eq.3b) F = 6 – E
Solution
Substitute 9 – E for C (from eq.4b), 6 – E for F (from eq.3b), 3 + E for B (from eq.2a), and 6 for D in eq.5: 9 – E + 6 – E = 3 + E + 6 which becomes 15 – 2×E = E + 9 In the equation above, add 2×E to each side, and subtract 9 from each side: 15 – 2×E + 2×E – 9 = E + 9 + 2×E – 9 which makes 6 = 3×E Divide both sides by 3: 6 ÷ 3 = 3×E ÷ 3 which means 2 = E making B = 3 + E = 3 + 2 = 5 (from eq.2a) C = 9 – E = 9 – 2 = 7 (from eq.4b) F = 6 – E = 6 – 2 = 4 (from eq.3b) and ABCDEF = 357624