# Pilsen circus

The arrival of the circus in Pilsen was seen in the morning at 08:00 by a citizen of the city. He passed this information on 08:15 to three other residents of the city. Each of these three people then informed the other three residents at 08:30, and again at 08:45, they reported the arrival of the circus every three to other uninformed residents. This chain continued the same way every other quarter of an hour. What time did all residents of Pilsen first know about the arrival of the circus when each person was only informed once? Plzen = 171,707 inhabitants.

Result

t = 11:00

#### Solution:

$s = 171707 \ \\ \ \\ q = 3 \ \\ a_{ 1 } = 1 \ \\ \ \\ \ \\ s = a_{ 1 } \cdot \ \dfrac{ q^n-1 }{ q-1 } \ \\ \ \\ s/a_{ 1 } \cdot \ (q-1)+1 = q^n \ \\ \ \\ \ln s/a_{ 1 } \cdot \ (q-1)+1 = n \ln q \ \\ \ \\ n = \dfrac{ \ln (s/a_{ 1 } \cdot \ (q-1)+1) }{ \ln(q) } = \dfrac{ \ln (171707/1 \cdot \ (3-1)+1) }{ \ln(3) } \doteq 11.6025 \ \\ \ \\ t_{ 1 } = 8.00 + 0.25 \cdot \ (n-1) = 8.00 + 0.25 \cdot \ (11.6025-1) \doteq 10.6506 \ \\ \ \\ t = \lceil t_{ 1 } \rceil = \lceil 10.6506 \rceil = 11 = 11:00$

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