Playing with Cards

$\text{A card remains facing upward at the end if it is flipped by an odd number of persons. }\newline$

$\textbf{For n }\bf{ \leq } \textbf{ 200 }\newline$

$\text{Card }n\text{ is flipped by every person whose number is the divisor of }n\text{. For example, Card }8\text{ is flipped by person }$ $1, 2, 4 \text{ and }8.\text{ Card }n \text{ will be facing upward if number of divisors of }n \text{ will be an odd number. }\newline \text{Let number of divisors of }n \text{ be denoted by }d(n)\text{ and }n\text{ has the prime factorization as : }n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_m^{a_m}\text{, then }\newline d(n) = (a_1+1)(a_2+1)(a_3+1)\cdots(a_m+1)\newline \text{Now }d(n)\text{ will be odd only if all the factors }(a_i+1) \text{ are odd i.e. all }a_i's \text{ must be even. If all the }a_i's \text{ are even then }n \text{ is a perfect square. }\newline\text{So Card }n \text{ will be facing upward if }n\text{ is a perfect square. All square numbers from }1\text{ to }200 \text{ are }1^2, 2^2, 3^2, \cdots , 14^2. \newline\text{Hence there are }14 \text{ upward facing cards in the card range }1\text{ to }200.$

$\textbf{For 201 } \leq \textbf{ n } \leq \textbf{ 380 }$

$\text{Since there are no more than }200\text{ persons, the number of persons flipping Card }n \text{ is }1\text{ less than the number of divisors of }n.\newline\text{So Card }n \text{ will be facing upward if }d(n) \text{ is an even number. As shown in upper part that }d(n)\text{ is odd for square numbers and hence even for non }\newline\text{square numbers. Number of square numbers in this range is }5\,(15^2, 16^2, \cdots, 19^2)\newline\text{So the number of non square numbers in this range is }(380-201+1)-5 = 175.\newline\text{Hence there are }175 \text{ upward facing cards in the card range }201\text{ to }380.$

$\text{ So there are 175 + 14 = 189 upward facing cards at the end.}$