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{\small “‰‚ ÷‰‘ô ¡‰À�} \\[3mm]
{\Large \tookhali ¢ø¤ùı �õ‰�¥ª‰ü �󉪉³‰ƒ‰‘¢ õ‰³‰ƒ‰�—‰Â} \\[3mm]
{\large \sayedar �¥õ‰�ö •‰‘ş‰‘ö—‰Âô ¢¤§ �ó‰Ú‰�¤ş‰µ‰İû‰‘} \\[7mm]
{ ª‰€‰±‰‚ 42 ¡‰Â¢�¢ 6731 } \\[-6mm]
{õ‰À¤§: õ‰½‰Ş‰À õ‰ú‰Àş‰‘ö \hfill ø쉴: 5 ¨‰‘䉴}
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\begin{enumerate}
\item )4 ÷‰Ş‰Âù( õ‰Æ‰ÿ‰Ü‰‚ı ¤÷‰Ù�õ‰ƒ‰Ãı 󉃉Ɖµ‰ü )\InE{}List Coloring\EnE{}( “‰‚ �ş‰ß ¬‰�¤– —‰ã‰Âş‰Ó õ‰üª‰�¢:
\begin{quote}
{\bf ø¤ø¢ı: } ş‰× ğ‰Â�é ¨‰‘¢ùı \InE{}$G$\EnE{}, ş‰× õ‰¹‰Ş‰�䉂 õ‰‘÷‰€‰À \InE{}$A$\EnE{} �¥ ¤÷‰Ùû‰‘ 
              ø “‰Â�ı û‰Â ş‰× �¥ ¤�‰�§ \InE{}$G$\EnE{}, ş‰× 󉃉Ɖ´ �¥ ¤÷‰Ùû‰‘ \\
{\bf ¨‰\hamze ��ñ: } �ş‰‘ õ‰ü—‰��ö õ‰¹‰Ş‰�䉂ı ¤�‰�§ \InE{}$G$\EnE{} ¤� “‰‘ ¤÷‰Ùû‰‘ı \InE{}$A$\EnE{} “‰‚ ¬‰�¤—‰ü ¤÷‰Ù î‰Â¢
               ¤÷‰Ù û‰Â ¤\hamze �§ �¥ 󉃉Ɖµ‰Ç �÷‰µ‰¿‰‘’ ª‰Àù  “‰‘ª‰À ø û‰Â ¢ø ¤\hamze �§ õ‰¹‰‘ø¤ı
              ¤÷‰Ùû‰‘ı õ‰µ‰Ô‰‘ø– ¢�ª‰µ‰‚ “‰‘ª‰€‰À?
\end{quote}
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\begin{enumerate}
\item ™‰‘“‰´ ‰ƒ‰À õ‰Æ‰ÿ‰Ü‰‚ı ¤÷‰Ù�õ‰ƒ‰Ãı 󉃉Ɖµ‰ü ¢¤ Ÿ‰‘󉵉ü  û‰Â ş‰× �¥ ¤÷‰Ùû‰‘ ¢¤ Ÿ‰À�‰Â \InE{}$k$\EnE{} —‰‘
      �¥ 󉃉Ɖ´û‰‘ �õ‰Àù “‰‘ª‰À ø û‰Â ş‰× �¥  󉃉Ɖ´û‰‘  {\em ¢ì‰ƒ‰Ö‰\nasb ‘} \InE{}$k$\EnE{} 䉀‰Ê‰Â ¢�ª‰µ‰‚ “‰‘ª‰À
      쉑“‰Û Ÿ‰Û ¢¤ ¥õ‰‘ö �‰€‰À›‰Ş‰Ü‰‚�ı �¨‰´.
\item ™‰‘“‰´ ‰ƒ‰À õ‰Æ‰ÿ‰Ü‰‚ı ¤÷‰Ù�õ‰ƒ‰Ãı 󉃉Ɖµ‰ü ¢¤ Ÿ‰‘󉵉ü  û‰Â ş‰× �¥ 󉃉Ɖ´û‰‘ Ÿ‰À�‰Â ¢ø 䉀‰Ê‰Â ¢�ª‰µ‰‚
      “‰‘ª‰€‰À, 쉑“‰Û Ÿ‰Û ¢¤ ¥õ‰‘ö �‰€‰À›‰Ş‰Ü‰‚�ı �¨‰´.
\item ™‰‘“‰´ ‰ƒ‰À õ‰Æ‰ÿ‰Ü‰‚ı ¤÷‰Ù�õ‰ƒ‰Ãı 󉃉Ɖµ‰ü ¢¤ Ÿ‰‘󉵉ü  û‰Â ş‰× �¥ ¤÷‰Ùû‰‘ ¢¤ Ÿ‰À�‰Â ¢ø—‰‘
      �¥ 󉃉Ɖ´û‰‘ �õ‰Àù “‰‘ª‰À, 쉑“‰Û Ÿ‰Û ¢¤ ¥õ‰‘ö �‰€‰À›‰Ş‰Ü‰‚�ı �¨‰´.
\item ™‰‘“‰´ ‰ƒ‰À õ‰Æ‰ÿ‰Ü‰‚ı ¤÷‰Ù�õ‰ƒ‰Ãı 󉃉Ɖµ‰ü ¢¤ Ÿ‰‘󉵉ü  û‰Â ş‰× �¥ 󉃉Ɖ´û‰‘ Ÿ‰À�‰Â 3 䉀‰Ê‰Â
      ¢�ª‰µ‰‚ “‰‘ª‰€‰À \InE{}NP-Complete\EnE{} �¨‰´.
\end{enumerate}


\item )2 ÷‰Ş‰Âù( õ‰Æ‰ÿ‰Ü‰‚ı ¥õ‰‘ö“‰€‰Àı ¤û‰‘ “‰‘ ¤�“‰Î‰‚ı •‰ƒ‰Ç÷‰ƒ‰‘¥ı \InE{}OR\EnE{} “‰‚ ¬‰�¤– ¥ş‰Â �¨‰´:
\begin{quote}
{\bf ø¤ø¢ı: } ş‰× õ‰¹‰Ş‰�䉂ı \InE{}$T$\EnE{} �¥ ¤û‰‘ş‰ü  ¥õ‰‘ö �÷‰¹‰‘ô û‰Â î‰À�ô �¥ �öû‰‘ ø�Ÿ‰À
              �¨‰´, ş‰× ğ‰Â�é ›‰ú‰´¢�¤ “‰Àøö ¢ø¤ õ‰‘÷‰€‰À \InE{}$G$\EnE{} “‰‘ õ‰¹‰Ş‰�䉂ı ¤�‰�§ \InE{}$T$\EnE{} 
              ş‰× ¤�“‰Î‰‚ı •‰ƒ‰Ç÷‰ƒ‰‘¥ı “‰ƒ‰ß ¤û‰‘ ¤� õ‰È‰¿‰É õ‰ü‰À,
              ş‰× ä‰À¢ ¬‰½‰ƒ‰¼ \InE{}$d_t$\EnE{} “‰‚ 䉀‰��ö ¥õ‰‘ö •‰‘ş‰‘ö )\InE{}deadline\EnE{}( “‰Â�ı û‰Â 
              ¤ \InE{}$t\in T$\EnE{}. \\
{\bf ¨‰\hamze ��ñ: } �ş‰‘ õ‰ü—‰��ö ¤û‰‘ı \InE{}$T$\EnE{} ¤� “‰Â ¤øı ş‰× •‰Â¢�¥÷‰Àù “‰‚ ğ‰�÷‰‚�ı ¥õ‰‘ö“‰€‰Àı 
              î‰Â¢  쉱‰Û �¥ �›‰Â�ı û‰Â ş‰× �¥ ¤û‰‘ş‰ü  •‰ƒ‰Ç÷‰ƒ‰‘¥ ¢�ª‰µ‰‚ “‰‘ª‰€‰À, 
              Ÿ‰À�ì‰Û ş‰Ø‰ü �¥ •‰ƒ‰Ç÷‰ƒ‰‘¥û‰‘ı �ö �›‰Â� ª‰Àù “‰‘ª‰€‰À ø û‰Â î‰À�ô �¥ ¤û‰‘ ÷‰ƒ‰Ã
              쉱‰Û �¥ ¥õ‰‘ö •‰‘ş‰‘ö õ‰Â“‰�ω‚ —‰Ş‰‘ô ª‰Àù “‰‘ª‰À.

              “‰‚ 䉱‰‘¤– ¢ì‰ƒ‰Õ—‰Â, õ‰ü¡‰��û‰ƒ‰İ —‰‘“‰â ş‰× “‰‚ ş‰× 
              \InE{}$\sigma: T \mapsto \{0, 1, \ldots\}$\EnE{} ¤� “‰‚ ğ‰�÷‰‚�ı •‰ƒ‰À� ‰ƒ‰İ 
              “‰Â�ı û‰Â ¤ \InE{}$t\in T$\EnE{}, \InE{}$\sigma(t) < d_t$\EnE{} ø 
              �ğ‰Â ¢¤›‰‚ı ø¤ø¢ı \InE{}$t$\EnE{} ¢¤ ğ‰Â�é \InE{}$G$\EnE{} �¥ ¬‰Ô‰Â “‰ƒ‰È‰µ‰Â “‰‘ª‰À, Ÿ‰À�ì‰Û ş‰× 
              ¤ \InE{}$t'\in T$\EnE{} ø›‰�¢ ¢�ª‰µ‰‚ “‰‘ª‰À  \InE{}$(t',t) \in E(G)$\EnE{}
              ø \InE{}$\sigma(t') < \sigma(t)$\EnE{}.
\end{quote}
™‰‘“‰´ ‰ƒ‰À  õ‰Æ‰ÿ‰Ü‰‚ı ê‰�ë \InE{}NP-Complete\EnE{} “‰‚ õ‰Ô‰ú‰�ô ì‰�ı �¨‰´. 
%)¤�û‰€‰Ş‰‘ş‰ü: õ‰Æ‰ÿ‰Ü‰‚ı \InE{}X3C\EnE{} ¤� “‰‚ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ —‰±‰Àş‰Û ‰ƒ‰À.(

\item )2 ÷‰Ş‰Âù( “‰‘¥ı \InE{}Misere Lasker Nim\EnE{} “‰‚ �ş‰ß ¬‰�¤– —‰ã‰Âş‰Ó õ‰üª‰�¢: 
¢¤ �“‰µ‰À�, �‰€‰À ¢¨‰µ‰‚  ¢¤ û‰Â ş‰× —‰ã‰À�¢ı ¨‰€‰Ú‰Âş‰Ãù ì‰Â�¤ ¢�¤¢, ¢�¤ş‰İ. 
¢¤ û‰Â Ÿ‰Â, “‰‘¥ı‰ü  ÷‰�“‰´ Ÿ‰Â “‰‘ �ø¨‰´ õ‰ü—‰��÷‰À û‰Â —‰ã‰À�¢ )õ‰·‰±‰µ‰ü( 
 “‰¿‰��û‰À �¥ ş‰× ¢¨‰µ‰‚ “‰Â¢�¤¢ ø ş‰‘ ş‰× ¢¨‰µ‰‚ ¤� “‰‚ ¢ø ¢¨‰µ‰‚ı ÷‰‘—‰ú‰ü —‰Ö‰Æ‰ƒ‰İ 
‰À. î‰Æ‰ü  �¡‰Âş‰ß Ÿ‰Â ¤� �÷‰¹‰‘ô ¢û‰À {\em “‰‘¥÷‰Àùı } “‰‘¥ı �¨‰´. ª‰Â¯ „¥ô 
ø ê‰ü “‰Â�ı “‰Â÷‰Àù ª‰Àö ÷‰Ô‰Â ¢øô ¢¤ �ş‰ß “‰‘¥ı ¤� “‰‚ ¢¨‰´ �ø¤ş‰À.

\item )1 ÷‰Ş‰Âù( —‰€‰ú‰‘ “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ —‰ã‰Âş‰Ó ™‰‘“‰´ ‰ƒ‰À õ‰¹‰Ş‰�á ¢ø “‰‘¥ı 
\InE{}$1\over2$\EnE{} õ‰ã‰‘¢ñ “‰‘ “‰‘¥ı 1 �¨‰´.

\item )1 ÷‰Ş‰Âù( ê‰Â­ ‰ƒ‰À  \InE{}$\cal A$\EnE{} õ‰¹‰Ş‰�䉂ı û‰Ş‰‚ı ¢÷‰±‰‘ó‰‚û‰‘ı ÷‰‘õ‰µ‰€‰‘û‰ü “‰‚ ª‰Ø‰Û 
\InE{}$(a_1, a_2, a_3, \ldots)$\EnE{}  “‰Â�ı û‰Â \InE{}$i>4$\EnE{}, \InE{}$a_i=a_{i-1}+a_{i-3}+a_{i-4}$\EnE{} “‰‘ª‰À.
õ‰¹‰Ş‰�\kasre á ¢ø ¢÷‰±‰‘ó‰‚ı \InE{}$(a_1, a_2, a_3, \ldots)$\EnE{} ø \InE{}$(b_1, b_2, b_3, \ldots)$\EnE{}, ¢÷‰±‰‘ó‰‚ı 
\InE{}$(a_1+b_1, a_2+b_2, a_3+b_3, \ldots)$\EnE{} �¨‰´. û‰Ş‰»‰€‰ƒ‰ß Ÿ‰‘¬‰Û®‰Â’ ä‰À¢ Ÿ‰Ö‰ƒ‰Ö‰ü \InE{}$c$\EnE{} ¢¤ ¢÷‰±‰‘ó‰‚ı 
\InE{}$(a_1, a_2, a_3, \ldots)$\EnE{}, ¢÷‰±‰‘ó‰‚ı \InE{}$(ca_1, ca_2, ca_3, \ldots)$\EnE{} �¨‰´. ™‰‘“‰´ ‰ƒ‰À  õ‰¹‰Ş‰�䉂ı
\InE{}$\cal A$\EnE{} “‰‘ �ş‰ß 䉪‰Ûû‰‘ —‰È‰Ø‰ƒ‰Û ş‰× ê‰Ì‰‘ı “‰Â¢�¤ı ¤øı õ‰ƒ‰À�ö ä‰À¢û‰‘ı Ÿ‰Ö‰ƒ‰Ö‰ü õ‰ü¢û‰À. 
\zamme “‰ã‰À �ş‰ß ê‰Ì‰‘ �‰€‰À �¨‰´?

\item )2 ÷‰Ş‰Âù( ê‰Â­ ‰ƒ‰À \InE{}$G$\EnE{} ş‰× ğ‰Â�é ¨‰‘¢ù ø \InE{}$E$\EnE{} õ‰¹‰Ş‰�䉂ı ş‰‘ñû‰‘ı \InE{}$G$\EnE{} “‰‘ª‰À. 
“‰‚ ¨‰‘¢ğ‰ü õ‰ü—‰��÷‰ƒ‰À —‰½‰Ö‰ƒ‰Õ ‰ƒ‰À  õ‰¹‰Ş‰�䉂ı û‰Ş‰‚ı ¥ş‰Âõ‰¹‰Ş‰�䉂û‰‘ı \InE{}$E$\EnE{} “‰‘ 䉪‰Û 
\InE{}$\Delta$\EnE{} )—‰Ô‰‘®‰Û õ‰µ‰Ö‰‘¤ö( —‰È‰Ø‰ƒ‰Û ş‰× ê‰Ì‰‘ı “‰Â¢�¤ı \InE{}$\cal V$\EnE{} ¤øı õ‰ƒ‰À�ö \InE{}$Z_2$\EnE{} õ‰ü¢û‰€‰À. 
%)®‰Â’ �¨‰Ø‰‘ó‰Â ¢¤ �ş‰ß ê‰Ì‰‘ı “‰Â¢�¤ı “‰‚ �ş‰ß ¬‰�¤– —‰ã‰Âş‰Ó õ‰üª‰�¢  “‰Â�ı û‰Â ¥ş‰Âõ‰¹‰Ş‰�䉂ı
%\InE{}$A\subseteq E$\EnE{}, \InE{}$0A = \emptyset$\EnE{} ø \InE{}$1A = A$\EnE{}(
\begin{enumerate}
\item ™‰‘“‰´ ‰ƒ‰À  û‰Ş‰‚ı ¥ş‰Âõ‰¹‰Ş‰�䉂û‰‘ş‰ü �¥ \InE{}$E$\EnE{}  õ‰µ‰€‰‘҉ “‰‘ ¥ş‰Âğ‰Â�éû‰‘ş‰ü �¥ \InE{}$G$\EnE{}
û‰Æ‰µ‰€‰À  ¢¤›‰‚ı û‰Â ¤\hamze �§ ¢¤ �öû‰‘ ¥øš �¨‰´, ş‰× ¥ş‰Âê‰Ì‰‘ı “‰Â¢�¤ı �¥ \InE{}$\cal V$\EnE{} ¤� —‰È‰Ø‰ƒ‰Û 
õ‰ü¢û‰€‰À. �ş‰ß ¥ş‰Âê‰Ì‰‘ ¤� \InE{}$\cal C$\EnE{} õ‰ü÷‰‘õ‰ƒ‰İ. ş‰× •‰‘ş‰‚ “‰Â�ı \InE{}$\cal C$\EnE{} •‰ƒ‰À� ‰ƒ‰À. 
“‰ã‰À �ş‰ß ê‰Ì‰‘ �‰€‰À �¨‰´? —‰ã‰À�¢ 䉀‰‘¬‰Â \InE{}$\cal C$\EnE{} �‰€‰À —‰‘¨‰´?
\item ™‰‘“‰´ ‰ƒ‰À  û‰Ş‰‚ı “‰Â©û‰‘ı ş‰‘ó‰ü \InE{}$G$\EnE{} —‰È‰Ø‰ƒ‰Û ş‰× ¥ş‰Âê‰Ì‰‘ı “‰Â¢�¤ı �¥ \InE{}$\cal V$\EnE{} ¤� 
õ‰ü¢û‰€‰À  �ö ¤� \InE{}$\cal B$\EnE{} õ‰ü÷‰‘õ‰ƒ‰İ. õ‰€‰Ñ‰�¤ �¥ ş‰× “‰Â© ş‰‘ó‰ü \InE{}$G$\EnE{}, õ‰¹‰Ş‰�䉂ı ş‰‘ñû‰‘ş‰ü 
�¥ \InE{}$G$\EnE{} �¨‰´  ş‰× ¨‰Â �öû‰‘ ¢¤ ş‰× õ‰¹‰Ş‰�䉂ı \InE{}$S$\EnE{}, ø ¨‰Â ¢ş‰Ú‰Âª‰‘ö ¢¤ õ‰¹‰Ş‰�䉂ı \InE{}$\overline{S}$\EnE{}
ì‰Â�¤ ¢�¤¢. )¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ û‰Â “‰Â© ş‰‘ó‰ü “‰‘ ş‰× õ‰¹‰Ş‰�䉂ı \InE{}$S$\EnE{} õ‰µ‰€‰‘҉ �¨‰´.(
ş‰× •‰‘ş‰‚ “‰Â�ı \InE{}$\cal B$\EnE{} •‰ƒ‰À� ‰ƒ‰À. “‰ã‰À �ş‰ß ê‰Ì‰‘ �‰€‰À �¨‰´? —‰ã‰À�¢ 䉀‰‘¬‰Â \InE{}$\cal B$\EnE{} 
�‰€‰À —‰‘¨‰´?
\end{enumerate}

\item )1 ÷‰Ş‰Âù( õ‰‘—‰Âş‰Å ¥ş‰Â  “‰‚ {\em õ‰‘—‰Âş‰Å ø�÷‰À¤õ‰�÷‰À} õ‰ã‰Âøé �¨‰´ ¤� ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À:
$$M=\left[
\begin{array}{cccc}
1&1&\cdots&1 \\
x_1&x_2&\cdots&x_n \\
x_1^2&x_2^2&\cdots&x_n^2 \\
\vdots&\vdots&&\vdots \\
x_1^{n-1}&x_2^{n-1}&\cdots&x_n^{n-1} 
\end{array}\right]$$
“‰Â�¨‰‘§ ì‰Ì‰ƒ‰‚ı �¨‰‘¨‰ü ›‰±‰Â õ‰ü¢�÷‰ƒ‰İ  û‰Â õ‰ã‰‘¢ó‰‚ı ¢¤›‰‚ı \InE{}$n$\EnE{} Ÿ‰À�‰Â \InE{}$n$\EnE{} ¤ş‰È‰‚ı Ÿ‰Ö‰ƒ‰Ö‰ü
¢�¤¢. “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß ì‰Ì‰ƒ‰‚ �™‰±‰‘– ‰ƒ‰À  �ğ‰Â \InE{}$x_i$\EnE{} û‰‘ ¢ø “‰‚ ¢ø õ‰µ‰Ô‰‘ø– 
“‰‘ª‰€‰À, õ‰‘—‰Âş‰Å \InE{}$M$\EnE{} ø�¤øö•‰Áş‰Â �¨‰´.

\item )1 ÷‰Ş‰Âù( ™‰‘“‰´ ‰ƒ‰À ª‰Â¯ „¥ô ø ê‰ü “‰Â�ı �ş‰ß  ¨‰Î‰Âû‰‘ı ş‰× õ‰‘—‰Âş‰Å \InE{}$m\times n$\EnE{} �¥
ä‰À¢û‰‘ı Ÿ‰Ö‰ƒ‰Ö‰ü õ‰Æ‰µ‰Ö‰Û ¡‰Î‰ü “‰‘ª‰€‰À �ş‰ß �¨‰´  ê‰Ì‰‘ı ¨‰µ‰�÷‰\kasre ü �ö “‰Â�“‰Â “‰‘ \InE{}$R^m$\EnE{} “‰‘ª‰À. 
)ş‰ã‰€‰ü ¨‰µ‰�öû‰‘ı �ö —‰È‰Ø‰ƒ‰Û ş‰× õ‰¹‰Ş‰�䉂ı õ‰�ó‰À “‰Â�ı \InE{}$R^m$\EnE{} ¢û‰€‰À.(

\item )1 ÷‰Ş‰Âù( ê‰Â­ ‰ƒ‰À 䉪‰Õ ş‰× ª‰±‰Ø‰‚ı õ‰Â—‰° ‰€‰Àù “‰‘ \InE{}$k$\EnE{} ø¤ø¢ı \InE{}$S(k)$\EnE{} ø 䉪‰Õ ş‰×
ª‰±‰Ø‰‚ı �¢è‰‘ôğ‰Â “‰‘ \InE{}$2k$\EnE{} ø¤ø¢ı \InE{}$M(k)$\EnE{} “‰‘ª‰À. û‰Ş‰»‰€‰ƒ‰ß ê‰Â­ ‰ƒ‰À ¢÷‰±‰‘ó‰‚�ı �¥ \InE{}$n$\EnE{}
ä‰À¢ ¢�¤ş‰İ  õ‰ü¡‰��û‰ƒ‰İ õ‰Â—‰±‰È‰‘ö ‰ƒ‰İ ø ®‰Ş‰€‰\nasb ‘ õ‰ü¢�÷‰ƒ‰İ  û‰Â ş‰× �¥ �ş‰ß �ä‰À�¢
¢¤ “‰‘¥ùı \InE{}$k$\EnE{} —‰‘ş‰ü ÷‰Æ‰±‰´ “‰‚ õ‰�ì‰ã‰ƒ‰´ õ‰Â—‰° ª‰Àùª‰‘ö ì‰Â�¤ ¢�¤÷‰À. ÷‰È‰‘ö ¢û‰ƒ‰À õ‰ü—‰��÷‰ƒ‰İ
\InE{}$n$\EnE{} ä‰À¢ ¤� “‰‘ 䉪‰Õ \InE{}$S(k)+2M(k)$\EnE{} õ‰Â—‰° ‰ƒ‰İ.

\item )2 ÷‰Ş‰Âù( ¢¤ õ‰�¤¢ û‰Â ş‰× �¥ õ‰��¤¢ ¥ş‰Â ¢¤ Ÿ‰À�‰Â 3 ¡‰Í —‰�®‰ƒ‰¼ ¢û‰ƒ‰À:
\begin{enumerate}
\item �ó‰Ú‰�¤ş‰µ‰İû‰‘ı \InE{}Las Vegas\EnE{} 
\item �ó‰Ú‰�¤ş‰µ‰İû‰‘ı \InE{}Monte Carlo\EnE{}
%\item �÷‰µ‰Ú‰Â�ñğ‰ƒ‰Âı “‰‚ ¤ø© \InE{}Monte Carlo\EnE{} 
\item ¤ø© \InE{}Iterative Deepening\EnE{}
\item ¤ø© \InE{}$\alpha$-$\beta$ pruning\EnE{}
\end{enumerate}

\item )2 ÷‰Ş‰Âù( õ‰ã‰Ş‰‘ı õ‰Â—‰° î‰Â¢ö 8 ä‰À¢ ¢¤ ş‰× ¬‰Ô‰½‰‚ı \InE{}$3\times3$\EnE{} “‰‘ ş‰× ¡‰‘÷‰‚ı ¡‰‘ó‰ü
¤� ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À. )¢¤ �ş‰ß õ‰ã‰Ş‰‘ ¢¤ û‰Â Ÿ‰Â õ‰ü—‰��÷‰ƒ‰İ ş‰× ä‰À¢ ¤� “‰‚ ş‰× ¡‰‘÷‰‚ı
¡‰‘ó‰ü õ‰¹‰‘ø¤© �÷‰µ‰Ö‰‘ñ ¢û‰ƒ‰İ.(:
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\vskip-12mm
\begin{enumerate}
\item ş‰× �ó‰Ú‰�¤ş‰µ‰İ “‰‚ ¤ø© —‰³‰‚ ÷‰�¤¢ı )\InE{}Hill Climbing\EnE{}( “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ •‰ƒ‰È‰€‰ú‰‘¢ 
‰ƒ‰À. �ş‰‘ �ó‰Ú‰�¤ş‰µ‰İ ª‰Ş‰‘ “‰Â�ı û‰Ş‰‚ı Ÿ‰‘ó‰´û‰‘ı �ø󉃉‚ õ‰ü—‰��÷‰À �ş‰ß õ‰ã‰Ş‰‘ ¤� Ÿ‰Û ‰À? 
“‰Â�ı �¢ä‰‘ı ¡‰�¢ ¢ó‰ƒ‰Û “‰ƒ‰‘ø¤ş‰À.
\item ê‰Â­ ‰ƒ‰À  õ‰ü¡‰��û‰ƒ‰İ �ş‰ß õ‰ã‰Ş‰‘ ¤� “‰‘ Ÿ‰À�ì‰Û —‰ã‰À�¢ Ÿ‰Â Ÿ‰Û ‰ƒ‰İ. ş‰× 
—‰‘“‰â \InE{}heuristic\EnE{} “‰Â�ı �¤¥ş‰‘“‰ü ꉑ¬‰Ü‰‚ı ø®‰ã‰ƒ‰´ ê‰ã‰Ü‰ü —‰‘ û‰Àé •‰ƒ‰È‰€‰ú‰‘¢ ‰ƒ‰À ø “‰‘ 
�¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß —‰‘“‰â, �ó‰Ú‰�¤ş‰µ‰İ \InE{}Best First Search\EnE{} ¤� “‰Â ¤øı ø®‰ã‰ƒ‰´ �ø󉃉‚�ı 
 ¢¤ ª‰Ø‰Û ê‰�ë ÷‰È‰‘ö ¢�¢ù ª‰Àù �¨‰´ �÷‰¹‰‘ô ¢û‰ƒ‰À. õ‰Â�Ÿ‰Û õ‰¿‰µ‰Ü‰Ó �›‰Â�ı �ó‰Ú‰�¤ş‰µ‰İ 
—‰‘ ¤¨‰ƒ‰Àö “‰‚ ›‰��’ ¤� ÷‰È‰‘ö ¢û‰ƒ‰À.
\end{enumerate}
\item )3 ÷‰Ş‰Âù( ê‰Â­ ‰ƒ‰À ş‰× ¨‰ƒ‰Æ‰µ‰İ õ‰³‰ƒ‰�—‰Âı �‰€‰À¤“‰Âù ¢¤ �¡‰µ‰ƒ‰‘¤ ¢�¤ş‰İ. �ş‰ß ¨‰ƒ‰Æ‰µ‰İ 
õ‰ü—‰��÷‰À “‰Â÷‰‘õ‰‚û‰‘ı õ‰Â“‰�¯ “‰‚ �‰€‰À ¤“‰Â  “‰Â ¤øı —‰Âõ‰ƒ‰€‰‘ñû‰‘ı õ‰µ‰Ô‰‘ø—‰ü 
÷‰È‰Æ‰µ‰‚�÷‰À ø “‰‘ �ş‰ß ¨‰ƒ‰Æ‰µ‰İ ¤ õ‰ü‰€‰À ¤� “‰‚ ¬‰�¤– û‰Ş‰Ãõ‰‘ö �›‰Â� ‰À. 
“‰Â÷‰‘õ‰‚û‰‘ş‰ü  “‰‚ ω�¤ õ‰��¥ı �›‰Â� õ‰üª‰�÷‰À, �¥ Ÿ‰‘ê‰Ñ‰‚û‰‘ı ›‰À�ğ‰‘÷‰‚�ı 
�¨‰µ‰Ô‰‘¢ù õ‰ü‰€‰À ø “‰€‰‘“‰Â�ş‰ß —‰À�¡‰Ü‰ü ¢¤ ¤ �öû‰‘ •‰Àş‰À ÷‰Ş‰ü�ş‰À. �󉱉µ‰‚ “‰Â�ı 
“‰Âì‰Â�¤ î‰Â¢ö �¤—‰±‰‘¯ “‰ƒ‰ß ¢ø ¤“‰Â, �ş‰ß ¨‰ƒ‰Æ‰µ‰İ �õ‰Ø‰‘ö �¨‰µ‰Ô‰‘¢ù �¥ Ÿ‰‘ê‰Ñ‰‚ı 
õ‰È‰µ‰Âí ø ¨‰Ş‰‘ê‰�¤ ¤� �¥ Ï‰Âş‰Õ ¢¨‰µ‰�¤�– ø —‰��“‰â ¥ş‰Â ¢¤ �¡‰µ‰ƒ‰‘¤ “‰Â÷‰‘õ‰‚÷‰�ş‰Å 
ì‰Â�¤ õ‰ü¢û‰À:
\begin{itemize}
\item {\InE{}\tt shmget(key, variable)\EnE{}}: �ş‰ß ¢¨‰µ‰�¤ ş‰Ø‰ü �¥ õ‰µ‰ç‰ƒ‰Âû‰‘ı “‰Â÷‰‘õ‰‚ õ‰‘÷‰€‰À 
\InE{}{\tt variable}\EnE{} ¤� “‰‚ ş‰× ¡‰‘÷‰‚ı õ‰È‰¿‰É �¥ Ÿ‰‘ê‰Ñ‰‚ı õ‰È‰µ‰Âí )Ÿ‰‘ê‰Ñ‰‚�ı  “‰ƒ‰ß 
“‰Â÷‰‘õ‰‚û‰‘ı õ‰¿‰µ‰Ü‰Ô‰ü  “‰Â  ¤øı �ş‰ß ¨‰ƒ‰Æ‰µ‰İ �›‰Â� õ‰üª‰�÷‰À õ‰È‰µ‰Âí �¨‰´.( 
÷‰Æ‰±‰´ õ‰ü¢û‰À. 
{\InE{}\tt key\EnE{}} ş‰× ä‰À¢ ¬‰½‰ƒ‰¼ �¨‰´  î‰À õ‰Â“‰�¯ “‰‚ ş‰× ¡‰‘÷‰‚ �¥ Ÿ‰‘ê‰Ñ‰‚ı õ‰È‰µ‰Âí 
“‰Â÷‰‘õ‰‚û‰‘  õ‰ü¡‰��û‰ƒ‰İ \InE{}{\tt variable}\EnE{} ¢¤ �ö ›‰‘ ì‰Â�¤ ğ‰ƒ‰Â¢ ¤� õ‰È‰¿‰É õ‰ü‰À. 
{\InE{}\tt variable\EnE{}} ÷‰ƒ‰Ã ş‰× õ‰µ‰ç‰ƒ‰Â �¥ ÷‰�á \InE{}Integer\EnE{} �¨‰´  õ‰ü¡‰��û‰ƒ‰İ �ö ¤� ¢¤ Ÿ‰‘ê‰Ñ‰‚ı
õ‰È‰µ‰Âí ì‰Â�¤ ¢û‰ƒ‰İ. 
•‰Å �¥ �›‰Â�ı �ş‰ß ¢¨‰µ‰�¤, õ‰µ‰ç‰ƒ‰Â \InE{}{\tt variable}\EnE{} ş‰× ¡‰‘÷‰‚ �¥ Ÿ‰‘ê‰Ñ‰‚ı õ‰È‰µ‰Âí
õ‰üª‰�¢ ø �ğ‰Â “‰Â÷‰‘õ‰‚ı ¢ş‰Ú‰Âı û‰İ õ‰µ‰ç‰ƒ‰Âı ¤� “‰‚ û‰Ş‰ƒ‰ß ¡‰‘÷‰‚ )“‰‘ û‰Ş‰ƒ‰ß î‰À( ÷‰Æ‰±‰´ ¢û‰À,
�ş‰ß ¢ø “‰Â÷‰‘õ‰‚ “‰‚ ω�¤ õ‰È‰µ‰Âí �¥ �ş‰ß õ‰µ‰ç‰ƒ‰Â �¨‰µ‰Ô‰‘¢ù ¡‰��û‰€‰À î‰Â¢.
\item \InE{}{\tt semop(key, value)}\EnE{}: �ş‰ß —‰‘“‰â “‰‚ õ‰Ö‰À�¤ ş‰× ¨‰Ş‰‘ê‰�¤  “‰‘ 
î‰À \InE{}{\tt key}\EnE{} ) ş‰× ä‰À¢ ¬‰½‰ƒ‰¼ �¨‰´( õ‰È‰¿‰É õ‰üª‰�¢, õ‰Ö‰À�¤ \InE{}{\tt value}\EnE{}  
ş‰× õ‰Ö‰À�¤ ¬‰½‰ƒ‰¼ õ‰·‰±‰´ ş‰‘ õ‰€‰Ô‰ü �¨‰´ ¤� �®‰‘ꉂ õ‰ü‰À.
õ‰ü¢�÷‰ƒ‰À  û‰Â ¨‰Ş‰‘ê‰�¤ ¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ ş‰× õ‰µ‰ç‰ƒ‰Â �¨‰´  õ‰ü—‰��÷‰À ş‰× ä‰À¢ ¬‰½‰ƒ‰¼ ÷‰‘õ‰€‰Ô‰ü ¤� 
¢¤ ¡‰�¢ ÷‰Ú‰‚ ¢�¤¢. ¢¤ ¬‰�¤—‰ü  ş‰× “‰Â÷‰‘õ‰‚ “‰¿‰��û‰À õ‰Ö‰À�¤ ş‰× ¨‰Ş‰‘ê‰�¤ ¤� —‰ç‰ƒ‰ƒ‰Â ¢û‰À, 
¨‰ƒ‰Æ‰µ‰İ 䉑õ‰Û �ş‰ß ¤ ¤� “‰Â�ı �ø �÷‰¹‰‘ô õ‰ü¢û‰À; �󉱉µ‰‚ —‰€‰ú‰‘ “‰‚ ª‰Âωü �ş‰ß ¤ �÷‰¹‰‘ô
õ‰üª‰�¢  •‰Å �¥ �÷‰¹‰‘ô �ş‰ß 䉪‰Û, õ‰Ö‰À�¤ ¨‰Ş‰‘ê‰�¤ û‰Ş‰»‰€‰‘ö ÷‰‘õ‰€‰Ô‰ü “‰Ş‰‘÷‰À.
—‰‘“‰â \InE{}{\tt semop}\EnE{} ¢¤ ¬‰�¤—‰ü  “‰µ‰��÷‰À õ‰Ö‰À�¤ \InE{}{\tt value}\EnE{} ¤� “‰‚ ¨‰Ş‰‘ê‰�¤ı  “‰‘
î‰À \InE{}{\tt key}\EnE{} õ‰È‰¿‰É õ‰üª‰�¢ �®‰‘ꉂ  ‰À ø ¢¤ ÷‰µ‰ƒ‰¹‰‚ı �ş‰ß ¤ õ‰Ö‰À�¤ ¨‰Ş‰‘ê‰�¤ õ‰€‰Ô‰ü
÷‰È‰�¢, õ‰Ö‰À�¤ \InE{}True\EnE{} ø ¢¤ 艃‰Â �ş‰ß ¬‰�¤– õ‰Ö‰À�¤ \InE{}False\EnE{} “‰Âõ‰üğ‰Â¢�÷‰À.
—‰�›‰‚ ‰ƒ‰À  �ş‰ß 䉪‰Û “‰‚ ¬‰�¤– 艃‰Â쉑“‰Û—‰Ö‰Æ‰ƒ‰İ )\InE{}atomic\EnE{}( �÷‰¹‰‘ô õ‰üª‰�¢; ş‰ã‰€‰ü õ‰Ş‰Ø‰ß
÷‰ƒ‰Æ‰´ ¢ø 䉪‰Û \InE{}{\tt semop}\EnE{} “‰Â ¤øı ş‰× ¨‰Ş‰‘ê‰�¤ ¢¤ ş‰× 󉽉щ‚ ¢¤ ¨‰ƒ‰Æ‰µ‰İ �÷‰¹‰‘ô ª‰�¢.
)�ğ‰Â ¢ø “‰Â÷‰‘õ‰‚ “‰‚ ω�¤ û‰Ş‰Ãõ‰‘ö —‰‘“‰â \InE{}{\tt semop}\EnE{}  ¤� ¬‰À� “‰Ã÷‰€‰À, ¤ õ‰Â“‰�¯ “‰‚
ş‰Ø‰ü �¥ �öû‰‘ ¥ø¢—‰Â �÷‰¹‰‘ô õ‰üª‰�¢ ø •‰Å �¥ �÷‰¹‰‘ô ¤ �øó‰ü ¤ ¢øõ‰ü û‰İ �÷‰¹‰‘ô
õ‰üª‰�¢.(
\item \InE{}{\tt newshm}\EnE{}: �ş‰ß —‰‘“‰â î‰À ş‰× ¡‰‘÷‰‚ �¥ Ÿ‰‘ê‰Ñ‰‚ı õ‰È‰µ‰Âí  —‰‘ ‰�ö ¢¤ ¨‰ƒ‰Æ‰µ‰İ
—‰�¨‰Í û‰ƒ‰º “‰Â÷‰‘õ‰‚�ı �¨‰µ‰Ô‰‘¢ù ÷‰È‰Àù �¨‰´ ¤� “‰Âõ‰üğ‰Â¢�÷‰À. 
)õ‰Æ‰Ü‰Ş‰\nasb ‘ û‰Â “‰‘¤  �ş‰ß —‰‘“‰â ¬‰À� ¥¢ù ª‰�¢ ş‰× õ‰Ö‰À�¤ ›‰Àş‰À ¤� “‰Âõ‰üğ‰Â¢�÷‰À.(
\item \InE{}{\tt newsem}\EnE{}: �ş‰ß —‰‘“‰â î‰À ş‰× ¨‰Ş‰‘ê‰�¤  —‰‘ ‰�ö ¢¤ ¨‰ƒ‰Æ‰µ‰İ —‰�¨‰Í û‰ƒ‰º 
“‰Â÷‰‘õ‰‚�ı �¨‰µ‰Ô‰‘¢ù ÷‰È‰Àù �¨‰´ ¤� “‰Âõ‰üğ‰Â¢�÷‰À. 
\end{itemize}
“‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß �õ‰Ø‰‘÷‰‘– õ‰ü¡‰��û‰ƒ‰İ ş‰× “‰‘¥ı ¢ø÷‰Ô‰Âùı “‰Â¢�ª‰µ‰ß ¨‰€‰Ú‰Âş‰Ãùû‰‘ 
“‰Â ¤øı �ş‰ß ¨‰ƒ‰Æ‰µ‰İ ωÂ�Ÿ‰ü ‰ƒ‰İ, “‰‚ ω�¤ı  ¢ø ¤“‰Â “‰µ‰��÷‰€‰À ¤øı ¢ø —‰Âõ‰ƒ‰€‰‘ñ 
õ‰¿‰µ‰Ü‰Ó “‰‘ û‰İ �ş‰ß “‰‘¥ı ¤� �÷‰¹‰‘ô ¢û‰€‰À. “‰Â�ı �ş‰ß õ‰€‰Ñ‰�¤ “‰‘ş‰À {\em ş‰×} “‰Â÷‰‘õ‰‚ 
“‰€‰�ş‰Æ‰ƒ‰İ  ¤“‰Â�ö õ‰¿‰µ‰Ü‰Ó “‰‘ �›‰Â�ı �ö “‰µ‰��÷‰€‰À “‰‘ û‰İ “‰‘¥ı ‰€‰À.
“‰Â÷‰‘õ‰‚ “‰‘ş‰À “‰‚ �ş‰ß ¬‰�¤– 䉪‰Û ‰À: 
�ğ‰Â ş‰× ¤“‰Â �ö ¤� �›‰Â� î‰Â¢, õ‰€‰µ‰Ñ‰Â ª‰�¢ —‰‘ ¤“‰Â ¢øô û‰İ û‰Ş‰ƒ‰ß “‰Â÷‰‘õ‰‚ ¤� �›‰Â�
‰À. •‰Å �¥ �ş‰ß, “‰ƒ‰ß �ş‰ß ¢ø ÷‰Ô‰Â �¤—‰±‰‘¯ “‰Âì‰Â�¤ õ‰üª‰�¢ ø õ‰ü—‰��÷‰€‰À “‰‘ û‰İ “‰‘¥ı 
‰€‰À. “‰‘¥ı “‰‚ �ş‰ß ¬‰�¤– �¨‰´  ¢¤ �“‰µ‰À�ı “‰‘¥ı, ş‰× ä‰À¢ ¬‰½‰ƒ‰¼ \InE{}$n$\EnE{}  ÷‰È‰‘ö¢û‰€‰Àùı
—‰ã‰À�¢ ¨‰€‰Ú‰Âş‰Ãùû‰‘ �¨‰´ �¥ “‰‘¥ı‰ü  “‰Â÷‰‘õ‰‚ ¤� �øñ �›‰Â� î‰Â¢ù �¨‰´  •‰Â¨‰ƒ‰Àù 
õ‰üª‰�¢. ¨‰³‰Å ¢¤ û‰Â õ‰ÂŸ‰Ü‰‚ �¥ ş‰Ø‰ü �¥ ¢ø “‰‘¥ıî‰ß )“‰‚ ¬‰�¤– ş‰× ¢¤ õ‰ƒ‰‘ö( ş‰× ä‰À¢
) õ‰ü—‰��÷‰À 1 ş‰‘ 2 “‰‘ª‰À( •‰Â¨‰ƒ‰Àù õ‰üª‰�¢ ø “‰‚ �ş‰ß —‰ã‰À�¢ �¥ ¨‰€‰Ú‰Âş‰Ãùû‰‘ “‰Â¢�ª‰µ‰‚
õ‰üª‰�¢. ¢¤ û‰Â õ‰ÂŸ‰Ü‰‚ —‰ã‰À�¢ ¨‰€‰Ú‰Âş‰Ãùû‰‘ı “‰‘ì‰üõ‰‘÷‰Àù “‰Â�ı û‰Â ¢ø ÷‰Ô‰Â ÷‰�ª‰µ‰‚ õ‰üª‰�¢.
ø쉵‰ü  û‰Ş‰‚ı ¨‰€‰Ú‰Âş‰Ãùû‰‘ “‰Â¢�ª‰µ‰‚ ª‰Àù “‰‘ª‰€‰À, “‰‘¥ı •‰‘ş‰‘ö õ‰üş‰‘“‰À ø “‰‘¥ı‰ü
 �¡‰Âş‰ß ¨‰€‰Ú‰Âş‰Ãù ¤� “‰Â¢�ª‰µ‰‚ �¨‰´ “‰Â÷‰Àù õ‰½‰Æ‰�’ õ‰üª‰�¢. ¢¤ �ş‰ß û‰€‰Ú‰‘ô “‰Â÷‰‘õ‰‚ 
•‰ƒ‰ç‰‘ô \InE{}You won\EnE{} )“‰Â�ı “‰Â÷‰Àù( ş‰‘ \InE{}You lose\EnE{} )“‰Â�ı “‰‘¥÷‰Àù( ¤� �‰‘” õ‰ü‰À ø �¥ “‰Â÷‰‘õ‰‚
¡‰‘¤š õ‰üª‰�¢.

“‰Â÷‰‘õ‰‚�ı “‰€‰�ş‰Æ‰ƒ‰À  䉪‰Û ê‰�ë ¤� �÷‰¹‰‘ô ¢û‰À. —‰�›‰‚ ¢�ª‰µ‰‚ “‰‘ª‰ƒ‰À  õ‰Ş‰Ø‰ß �¨‰´ 
¢¤ ş‰× 󉽉щ‚ “‰ƒ‰È‰µ‰Â �¥ ¢ø ÷‰Ô‰Â “‰Â÷‰‘õ‰‚ı ª‰Ş‰‘ ¤� �›‰Â� ‰€‰À. )¢¤ �ş‰ß ¬‰�¤– “‰‘ş‰À
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ª‰Ş‰‘( ÷‰Ş‰ü—‰��÷‰À �¥ Ÿ‰‘ê‰Ñ‰‚ı õ‰È‰µ‰Âí ø ¨‰Ş‰‘ê‰�¤ı  “‰Â÷‰‘õ‰‚ı ª‰Ş‰‘ �¥ �ö �¨‰µ‰Ô‰‘¢ù õ‰ü‰À 
�¨‰µ‰Ô‰‘¢ù ‰À. û‰Ş‰»‰€‰ƒ‰ß
ê‰Â­ ‰ƒ‰À  ¤“‰Â�ö û‰ƒ‰ºğ‰‘ù �ω…䉑– ÷‰‘¢¤¨‰´ “‰‚ “‰Â÷‰‘õ‰‚ı ª‰Ş‰‘ ÷‰Ş‰ü¢û‰€‰À. )õ‰·‰\nasb …
¢¤ ¬‰�¤—‰ü  ş‰× ¨‰€‰Ú‰Âş‰Ãù “‰‘ì‰ü õ‰‘÷‰Àù “‰�¢, ¤“‰Â ¨‰ã‰ü ÷‰Ş‰ü‰À  ¢ø ¨‰€‰Ú‰Âş‰Ãù 
“‰Â¢�¤¢.(
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\leftline{\small 'õ‰�ê‰Õ “‰‘ª‰ƒ‰À"}
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\item û‰Â õ‰À�¤ �ó‰Ø‰µ‰Âş‰Ø‰ü  —‰€‰ú‰‘ �¥ õ‰Ö‰‘øõ‰´û‰‘ —‰È‰Ø‰ƒ‰Û ª‰Àù �¨‰´ ¤� õ‰ü—‰��ö “‰‘ 
�¨‰µ‰Ô‰‘¢ù �¥ ş‰× ğ‰Â�é ›‰ú‰´¢�¤ ÷‰È‰‘ö ¢�¢; “‰‚ �ş‰ß ¬‰�¤–  û‰Â ş‰‘ñ �¥ ğ‰Â�é 
÷‰È‰‘ö¢û‰€‰Àùı ş‰× õ‰Ö‰‘øõ‰´ �¥ õ‰À�¤ ø û‰Â ¤\hamze �§ ğ‰Â�é ÷‰È‰‘ö¢û‰€‰Àùı ş‰× õ‰½‰Û �—‰Ê‰‘ñ 
�‰€‰À ¨‰ƒ‰İ �¨‰´. “‰Â�ı õ‰·‰‘ñ õ‰ü—‰��÷‰ƒ‰À ğ‰Â�é ¥ş‰Â ¤� ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À:
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“‰‚ û‰Â ¤\hamze �§ \InE{}$x$\EnE{} �¥ �ş‰ß ğ‰Â�é, ş‰× ä‰À¢ Ÿ‰Ö‰ƒ‰Ö‰ü \InE{}$v_x$\EnE{} ÷‰Æ‰±‰´ ¢�¢ù õ‰üª‰�¢  ÷‰È‰‘ö¢û‰€‰Àùı 
õ‰Ö‰À�¤ ø󉵉‘¦ ÷‰Ö‰Î‰‚ı �—‰Ê‰‘ñ õ‰µ‰€‰‘҉ “‰‘ \InE{}$x$\EnE{} �¨‰´. û‰Ş‰»‰€‰ƒ‰ß “‰‚ û‰Â ş‰‘ñ \InE{}$e$\EnE{} �¥ ğ‰Â�é, 
ş‰× ä‰À¢ \InE{}$I_e$\EnE{} ÷‰Æ‰±‰´ ¢�¢ù õ‰üª‰�¢  ÷‰È‰‘ö¢û‰€‰Àùı õ‰Ö‰À�¤ ›‰Âş‰‘÷‰ü �¨‰´  
�¥ õ‰Ö‰‘øõ‰´ õ‰Â“‰�¯ “‰‚ ş‰‘ñ \InE{}$e$\EnE{} )¢¤ ›‰ú‰µ‰ü  —‰�¨‰Í ş‰‘ñ \InE{}$e$\EnE{} õ‰È‰¿‰É ª‰Àù �¨‰´( õ‰üğ‰Á¤¢. 
)�ş‰ß õ‰Ö‰À�¤ õ‰ü—‰��÷‰À õ‰€‰Ô‰ü û‰İ “‰‘ª‰À.( 
“‰Â �¨‰‘§ 쉑÷‰�ö ‰Âª‰Ó, “‰Â�ı û‰Â ¤\hamze �§ \InE{}$x$\EnE{} �¥ ğ‰Â�é, õ‰¹‰Ş‰�á ›‰Âş‰‘\kasre ö ş‰‘ñû‰‘ş‰ü  
“‰‚ \InE{}$x$\EnE{} ø�¤¢ õ‰üª‰�÷‰À, “‰Â�“‰Â “‰‘ õ‰¹‰Ş‰�\kasre á ›‰Âş‰‘\kasre ö ş‰‘ñû‰‘ş‰ü �¨‰´  �¥ \InE{}$x$\EnE{} ¡‰‘¤š
õ‰üª‰�÷‰À. “‰‚ “‰ƒ‰‘ö ¢ş‰Ú‰Â, ›‰Âş‰‘öû‰‘ ¢¤ �¬‰Û “‰Ö‰‘ı ›‰Âş‰‘ö )\InE{}Flow Conservation\EnE{}( ¬‰Àë
õ‰ü‰€‰À. 䉅øù “‰Â�ş‰ß, “‰Â �¨‰‘§ 쉑÷‰�ö �û‰İ, “‰Â�ı û‰Â ş‰‘\kasre ñ \InE{}$e=xy$\EnE{} �¥ ğ‰Â�é, ş‰× ä‰À¢
Ÿ‰Ö‰ƒ‰Ö‰ü õ‰·‰±‰´ \InE{}$r_e$\EnE{} ø›‰�¢ ¢�¤¢  \InE{}$v_x-v_y=r_eI_e$\EnE{} �¨‰´.

�ş‰‘ “‰‘ —‰�›‰‚ “‰‚ ì‰��÷‰ƒ‰ß ê‰�ë ø “‰‘ õ‰Ô‰Âø­ “‰�¢ö õ‰Ö‰À�¤ \InE{}$r_e$\EnE{} û‰‘, õ‰ü—‰��ö õ‰Ö‰À�¤
ø󉵉‘¦ ¤�‰�§ ø ›‰Âş‰‘ö ş‰‘ñû‰‘ ¤� õ‰È‰¿‰É î‰Â¢? �ğ‰Â �ş‰ß ¤ õ‰Ş‰Ø‰ß ÷‰ƒ‰Æ‰´, õ‰Ö‰À�¤ ø󉵉‘¦
�‰€‰À—‰‘ �¥ ¤�‰�§ “‰‘ş‰À õ‰È‰¿‰É ª‰�¢ —‰‘ �ş‰ß ¤ �õ‰Ø‰‘ö•‰Áş‰Â ª‰�¢? )ê‰Â­ ‰ƒ‰À  ğ‰Â�é
û‰Ş‰±‰€‰À �¨‰´.(