                             .

                             1.
      ணࠬ᪠ .

                             2.
     㣨 ᫮ 室  ணࠬ, 믮 
 ⢨,    RENUM 몠 ASIC.

                             4.
      (1,1)  ந譠   ப,     
ᤥ ᢮ । 室 (誠 㦥 ⮨  樨  (1,1).  
ᥤ 樨 - (1,2),  (2,2), (2,1) - 먣  饣
⥪騩 室 ( ਢ    먣    (1,1).  樨
(3,1),(1,3),(3,3) - ⠪ ந ( 室   ਢ 
먣 ,  ப, 騩 ᫥騩 室, 먣뢠).
      樨 ,  砥 ।,  -
樨   묨 न⠬  ந묨   -
稭饣,    ⠫  樨  -  먣   .  
⢨⥫쭮 ⠪:  ந譮 樨   묨 -
⠬ ᫥  ⨬ 室    ,  
    न ⭠ (먣).   樨   
   ⭮ न⮩  ᥣ     -
   묨  न⠬  (ந    饣
। 室).   ᫥  (1,1)   
न,   祫, 稭騩  樨,     
न  ⭠,  ᥣ ᬮ ᤥ ⠪,  ⮡ ᫥ 
 室 誠 뫠  樨   묨 न⠬,  
 殢 -  樨 (1,1).
     ⠪, ᫨ i  j ,    ப  
室 ண ᥣ ந뢠,  - 먣뢠.

                            5.
     ᫨ N  M   (..  ᫨ (N,M)=1),    -
⢮=N+M-1. ᫨  k=(N,M)<>1,   讣 אַ-
쭨  ।⠢  ꥤ k 窮,    
   אַ㣮쭨 ࠧ (N/k,M/k),  
N/k  M/k  . 砥, 
              ⢮=k*(N/k+M/k-1)=N+M-(M,N).
⠪, ᥣ ⢮=N+M-(M,N).

                            6.
     㤥  ⥪    ன (S,i),  S -
⢮ ,  ⠢  ,  । ⥪騬 室  i-
 (i ࠢ 1  2).
      ⮣,  ⮡  (S,1)  뫠 먣譮  ப 1,  ।
 権 ᫥  室 (S-a_1,2), ... ,(S-a_k,2) 
   ந譠  ப 2 (      
㤥 ॢ );  ᫨  樨 먣  ப 2,
 (S,1) - ந譠  ப 1 (    㯠,  
ࠢ நࠥ).
       ᫥ 室 ப 1 ⠫ (S',2).   -
  ⨬ 室  ⮩ 樨  ᬮਬ   -
稢  (S",1) 먣譮,  ந譮  ⠪, -
     業.  ᫨ 믮 ᫥ -
⨢,    (S",1)      室, -
㥬,  ..
     ᫨    殢 ।,    먣-
 ⥪ ,  頥  ।饬 室  ⠥
।,     饣 室  . .,   -
  樨 (L,1).

const l1=200;
      s=10;
var   a: array [0..l1,1..2] of byte; {  祩 a[L,i] ࠭
                                      ଠ,  먣뢠
                                          樨 (L,i)}
      b: array [1..2,0..s] of byte;  { ᨢ  室    }
                                     { ࢮ  ண ப }
                                     {  b[i,0] ࠭ ᫮   }
                                     {  室 ப i  }
      l,i,j:integer;

  Function  f(ba:Integer;No: Byte): Byte;
   Var i,p,r: Byte;      { ४ᨢ 㭪 ᫥ (L,i) }
   Begin                    { f=i, ᫨  (ba,i) 먣뢠 i }
     if Ba<0             { ᫥ 室  <0? }
     Then  f:= 3-No      { 먣 ப  ஬ 3-N }
     else if ba=0                {  = 0? }
          then begin             { 먣 No}
                  f:=No;
                  a[ba,no]:=no;
               end

          Else if a[ba,no]<>0      {  ⮩ 樨  㦥 뫨 }
               Then F := a[ba,no]  {   먣뢠 a[ba,no] }
               Else
                    Begin {      ᬠਢ }
                         r := 0;
                         {  㤥    室 }
                         For i := 1 to b[No,0] do
                            if ba-b[no,i]>=0 { ᫨ 室  }
                                             {    }
                            then r := r or F(Ba-B[No,i],3-No);
                         if r=0 then r:=no;
                         If (No and R)<>0 {  먣 室? }
                         Then p := No     { , (ba,i)=No }
                         Else p := 3-No;  { , (ba,i)=3-No }
                         A[Ba,No]:=p; {   ଠ }
                         f:=p;
                    End;
   end;

begin
     for i:=0 to l1 do
      for j:=1 to 2 do  A[i,j] := 0;
    write('s=');
    readln(b[1,0]);     { b[1,0] = ⢮ 室 ࢮ ப }

    for i:=1 to b[1,0] do
    begin
     write('a',i,'=');  { ᠬ 室 }
     Readln(b[1,i]);
    end;
     write('k=');
     Readln(b[2,0]);    { b[2,0] = ⢮ 室 ண ப }
     For j := 1 to b[2,0] do
     begin
       write('b',j,'=');
       Readln(b[2,j]);  {  室 }
     end;
repeat                  {   室  祭 L }
     write('L=');       { ᫮  }
     readln(l);


     if f(l,1)=1        { 맮 ४ᨢ 㭪樨-஢ન }
     then writeln('1- 먣ࠫ')
     else writeln('1- நࠫ');

     for i:=1 to b[1,0] do  { ᯥ⪠ १⮢  室 }
     if (l-b[1,i])>=0       {  砫쭮 樨 L }
     then writeln('b=',b[1,i],' winner=',a[l-b[1,i],2]);
      writeln;
until false;
end.

                             7.
      襭 筠     .

                             8.
     ᫨ n=1,   ⢥ ᫮ 稭  稢 
    㪢,  楯窠 ஥.
      ⮣,  ⮡  뫮 㯮冷 ᫮ A1, ..., An
  A  "楯",  室,  ⮡ ᮢ㯭 㪢, 
 稢 ᫮,  ᮢ  ᮢ㯭  㪢,  
  ᫮ 稭 (㣨 ᫮, ⮡  -
饩 㪢 ᫮ ᮮ⢥⢮ ⠪  砫쭠 㪢  -
- ᫮).  ,    筮 ᫮, 뢠 -
: AB, BC, CA, DE, ED. "" ந .
        楯窠  易  ன १  ᫮,   
ࠢ,   ᫮  A 稭. 쬥 ᫮ A1. 塞
  ⢠ A: A:=A\A1. 饬  A - ᫮ Ai, -
饥  ᫥ 㪢 ᫮ A1 (⠪ ᫮ ᥣ  -
,  ⠪   ⠥,  室 ᫮, 㯮⮥ ,
믮).  塞 Ai  A: A:=A\Ai, 饬  A - Aj,
稭饥    ᫥ 㪢 Ai  ..  த   ,
   ᫮ A1,  ... ,An    楯 (⮣ 
襭),    ,  A  ,     ᫮,
稭饣  ᫥ 㪢 x ᫥  ᫮  (,  
室  ᫮ ᫥,    㪢 x ᮢ  ࢮ
㪢 ࢮ ᫮).  諨 - 楯.
     ਬ: A=(AB,BC,CA,BD,DC,CB).
     楯窠1: AB,BC,CA.
     窠 : AB,BD,DC,CB,BC,CA.
        諨 楯 1.  ६  - ᫮
 A  ⢥ 砫쭮  । 楯窨   -
塞 ஥ ᫥饩 楯窨,  ..
       殢  ⢮ A ⠫ ,   砫-
   ᫮ A1,  ...  ,An ᯠ  k 楯祪.  筥 
᪫:
       楯窠 i  j ᮮ⢥⢥  ᫮ u  v -
,   u  v ᮢ  㪢.  楯窠 i  
i=BuC,   j=EvD ( B,C,D,E -  ᫥⥫쭮 ᫮).
  i  j 楯窠 㤥  
     EuCBvD.
(஢,  ⠪ ᪫ ४⭠!)
       楯窨, 稬 ᪮ 楯. ᫨  -
- 蠣 ,   ᪫   楯祪  -
,   ࠧ訬.
     ਬ 1.   ਢ  ।饬 ਬ   ᫮
㤥 2 楯窨:
     AB, BC, CA    
     BD, DC, CB.
       ᮢ  㪢  ᫮ BC  BD.  ᫥ ᪫
砥
     AB, BD, DC, CB, BC, CA.
     ਬ 2.  A=(ACA, BB).  楯窨: ACA  BB.   ᮢ-
騬 묨 㪢   ,  ࠧ訬.

     ਠ #2 襭 .
     ᫨   ⠢  楯窨 ,     A 
᫮  ࢮ 㪢,  ᮢ饩  ᫥ 㪢  ᫥
᫮ Aj 楯窨,  Aj "饯塞"    ⠥  㣮
室饥 ᫮.  ᫨   㤠,   "饯塞"  ।
᫥ ᫮ 楯窨  ..  ᫨  - 蠣 A ,  
 襭,   ᫨  - 蠣 楯窠 ⠫  ⮩,
  ࠧ訬.
        ᫮ A=(A1, ... ,An).  ᫮ ਯᠭ
 㭨  ,  ࠢ .   㯮冷祭  -
⠭ .
     窠 = A1
     A=A\A1; =1.
      (A  )  (楯窠  ) 
     
       ᫨  A  室饥 ᫮  ᮬ > 
        ਯ뢠  楯窥  ᫮,
          ᫮ 㤠塞  A
          =1
        "饯塞"  楯窨 ᫥ ᫮ Aj
             =j
             Aj ᨬ  A
     
     ᫨ A ,  楯窠 
         襭.

      ⮤ 뢠 "ॡ  ⮬" ("backtracking").

                             9.
     ।,   誠 (㤥 뢠  1)  -
 ⮫쪮 ࠢ,   ࠢ 誠 2 - ⮫쪮 ,  室 -
 . 襭   ࢮ砫쭮 ⠭ ᢮
 襭     ࠭祭.
        室  1.  1 먣뢠,  ᫨ ⢮
⮪  誠 d(1,2) (砫  ࠢ m-2)    
楫    k+1.  ⢨⥫쭮,    1  ⠪ 室,  
d(1,2) ⭮ k+1.    室  ᤥ 2 (ਬ,    L
⮪ ),  1 ᫥騬 室 ᬥ頥  k+1-L ⮪ -
, ,  ⠪  ࠧ,  ᫥  室  1 ﭨ d(1,2) ᭮
⭮ k+1.   ⠪  0 ⠪   k+1 楫,   㯨
,   d(1,2)=0  ।騩 室  室 1,  ..  2
நࠫ.  ᫨ (m-2) mod (k+1)=0,   1   室  2
ந뢠.
     ᫨ ࠧ 誠 室   ,  ⠪    ࠢ,  
ᥣ  먣뢠    誠,  ᫥ 室 ன d(1,2) ⭮
k+1.

                             10.
     ᫥⥫쭮 ந ⠪:  ᭠砫    0,  ⥬
  ᫥饥 ⢨:  㦥 ᠭ  ਯ뢠
ࠢ   0  1, 1  0, ..
                0 -> 01 -> 0110 -> 01101001 ->...
     ,  ।  ᫥⥫쭮  㤮⢮
᫮ .
     ᫨  ᬮ ⮣ ᤥ,   ⥫⢮:
     㤥 㬥஢  ᫥⥫쭮,  稭    .
⠪,  a(0)a(1)=01,  a(2)a(3)=10=a(1)a(0). ⬥⨬,  ᫥-
⥫쭮 ⮨  '௨稪' 01  10 ( a(2k)a(2k+1) 
 a(0)a(1),  a(1)a(0)).  १ ⭮ , 稭-
騩     ᮬ a(2k)   ஢  㫥  
.  ।,   M   ᬥ  १.
᫨  ⠪ १,    १ 쭮 . -
 ,  ᭠砫  砫 १ P, ⥬  
ᬥ  १ S1,S2  S3 쭮 , S1=S2=S3=
=S:
                       P S1 S2 S3.
     㤥   १ A १ IAI.
      4  ਠ:
     1) IPI=2k, ISI=2l.
     㤥  0*  01  1*  10.    ᨬ-
 ᫥⥫쭮 M 뢠 ⠪ ,   ࠭:
     M=0*1*1*0*1*0*0* ...
        P,    S     ᨬ 0*  1*;
砫 १ P 㤥   k  0*  1*,  S -  l
. ६ 窨   - ᫥⥫쭮 M  ⮣ 
.    P,S1,S2  S3  稬  १ P', S1',S2'
  S3'      訬.  १ S1',S2'  S3' 
묨,      ⠭ l.  稫 ⨢-
稥  ⢥ত  쭮  S.
     2) IPI=2k, ISI=2l+1.
     १ S1 稭 ⮬   ᮬ.   -
१ S1 S2 ⭠ ( ࠢ 4l+2),      ᫮ -
  .   S1 ⭠, ,  襭 魮, ।-
,   㫥  S1 , 祬 , ⮣  S2  -
  祬 㫥?! ⨢稥  ⥬,  S1=S2.
     3) IPI=2k+1, ISI=2l+1.
     १ S2  S3        稭 ⮬ 
 ᮬ.  㦤  ਢ  㭪
2.
     4) IPI=2k+1, ISI=2l.
     ᫥   १  P    a(2k) (ᯮ,
 㬥 ⮢ 稭  ).     a(2k) a(2k+1)
  01,   10,   ᥣ a(2k)<>a(2k+1).   
S1,S2  S3 , ᮮ⢥⢥, a(2k+1), a(2k+2l+1), a(2k+4l+1).
⮬  ᮢ S1=S2=S3 ᫥, 
     a(2k+1)=a(2k+2l+1)=a(2k+4l+1)<>a(2k)=a(2k+2l)=a(2k+4l).
     ⥫쭮, ⠪ 
       a(2k)a(2k+1)=a(2k+2l)a(2k+2l+1)=a(2k+4l)a(2k+4l+1),
   ,  騥 १ S1 S2 S3 稭
 樨 a(2k),  IPI=2k,  ISI=2l,    砩 㦥  ࠧ࠭  
㭪 1.

                             11.
     ᬮਬ ᭠砫 砩,  a, b, c, xi - ࠫ.
     1)   ᫠.
     ᫨ a<c,    ࠧ訬,  ᫨ a   [c,d],  
 襭.
      a>d.  쭮 ᫮ ᫠  㤥  ⮣,  
    ᫠  㤥      .    㭪
ceil(x)=k+1, ᫨ k<x<=k+1, k - 楫. ,   砥, ᫨
ࠧ ,  室 ceil(a/d) ᫠.
      k=ceil(a/d).  ᫨  쬥 k ᫠  d,  -
襭  a 㤥 p=k*d-a.   k ᫠   㬥-
   祬  d-c.  ⢮ 㬥蠥 ᫠ 
 n=ceil(p/(d-c)).  ᫨ n>k,  ࠧ   (᫠-
,    㬥,  ,  祬   , 㣨
᫮,  a div c = a div d,   a mod c <> 0,  a mod d <> 0). ᫨
n<=k,  L, L=p div (d-c), ᫠ 㬥蠥  (d-c),  ᫠-
 -  p mod (d-c), ⠫    (ࠢ묨
ᨬ쭮).
     ᫨ a,  b, c, xi - 楫,   ࠪ᪨  -
.
     2)   ᮬ⥫.
     ⮤ 襭 ⮩  祭 ᯮ짮   -
襭    宦 ᨬ쭮   饩 -
᫥⥫쭮   ᫥⥫쭮  (.    20
 "७ ᮮ⭮襭  ᪮ ணࠬ஢").
      㤥 ᬠਢ ⮫쪮 砩 a>d.
     稬 १ L 㯮冷祭  ⠭ ⢮  -
⥫ ᫠ A,   १ [c,d], ⢮ ⥫ 
ॢ蠥 2*SQRT(A) (᫨ a=f*g,   f<=SQRT(a), g>=SQRT(a), ᥣ
ࠧ  f -   SQRT(a),  ⮫쪮   ࠧ g).   L1 -
   L, a L2- ᨬ.  ᨢ S[1..k] -
ᥬ    ⠭   ⥫ ᫠ a (  ᠬ ᫮
a=S[k]), 稭  쭮   L.  ⢥ত 
᫥,  k<=2*SQRT(A)+1.
     㤥 ᪠    ᫥⥫쭮  -
⮢  ᨢ S,    ⢠ L  ᫮ a=S[k] ⠪,  
 ᫥騩  ᫥⥫쭮     -
騩  ⭮,  ਭ饥  L,   㫥 -
⮪.
      ᨢ B[1..k]  C[1..k].   B[i] 㤥  -
  ᫥饣      ᫥⥫쭮 쭮
,  饩  S[i]  S[k] (B[i]=0,  ᫨  a  S[i]  
),    C[i] 㤥 ࠭ ᫮ ⮢  ⮩ -
쭮 ᫥⥫쭮 (⬥⨬,    ᬠਢ  ⮫쪮
⠪ ᫥⥫쭮,   ⭮   ᫥-
 童 ᫥⥫쭮  ।騩 ਭ  L,
 ⮪   㫥).
     砫       C,  ஬ ᫥,  ᥬ -
- 讥 ᫮,  ਬ 2*A.  ᫨ C[i]=2*A,   㤥
,   ᫥⥫쭮 ⮢, 饩  S[i]  S[k],
  㦥.
     ᫥   A  B   室    
楯窨, 饩  ⢠ L  ᫮ a.
     for i:=1 to k-1 do C[k]:=2*A;  { ᫥⥫쭮  }
     C[k]:=1;      { ᫥⥫쭮    A }
     B[k]:=0;      { ।饣   }
       for i:=k-1 downto 1 do
         for j:=i+1 downto k do
           if (S[j] div S[i] >=L1) and   { ⭮ S[j]/S[i] }
              (S[j] div S[i] <=L2) and   { ਭ L,   }
              (S[j] mod S[i] =0)         { ⮪ 㫥 }
           then if C[i]>C[j]+1    { ᫨  㦥 }
                then begin   { ᥩ ᫥⥫쭮, }
                        { 饩  S[i]  S[k], , 祬 }
                  C[i]:=C[j]+1;   { ࠭ ,  - }
                  B[i]:=j         { ४㥬    }
                end;              { ᫥饣  }
       i:=1; min:=2*A; j:=0;
       while S[i]<=L2 do
       begin
         if C[i]< min
         then begin
           min:=C[i];    { ४㥬  }
           j:=i;         { ࠭塞  ࢮ  }
         end;
         i:=i+1
       end;
       if j=0
       then writeln('  ')
       else begin
              writeln(':');
              write(A[j]);
              for i:=1 to min-1 do
              begin
                P:=j;
                j:=B[j];
                write(A[j]/A[p]) { ᯥ⪠ ᮬ⥫ }
              end
       end
      砥, ᫨ a, c, d, xi - 楫,  ᨢ S 頥  -
浪 ⠭   ⥫ ᫠ A, 稭  -
   L.  - 筮.

                             12.
     . 襭   ⮬,     ᫮ ਮ-
᪨ ७ - 㣮 ᫮ ( 24).

                             13.
     蠥 筮   23 ( "७ ᮮ⭮-
  ᪮ ணࠬ஢").

                             14.
     ᫨ ᫥⥫쭮 B  C ᮢ,   ભ 㪢
। .
     ᫨ B  C ࠧ ஢   樨 (..   A  뫨
ભ ᥤ 㪢),  । ભ 㪢 ,
  ᫥⥫쭮  A  -    (ਬ:  A='ABCDB',   B='ACDB',
C='ABDB').
     - ⨬ .  ६  砫   ᫥-
⥫쭮⥩  B   C  ᮢ騥 .  稬 ᫥-
⥫쭮 B'  C'.  ਬ B='albce', C='alcde', B'='bc', C'='cd',
..    ᫥ ᨬ  B  C ᮢ. ᫨  ᬮ-
 ⠭ ᫮,  ண 砥 B'  C',   ᬮ-
 ⠭  A.
      B'=b1b2b3 ... bk
           C'=c1c2c3 ... ck.
     ᫨ b1=c2=b3=c4= ...               (*)
          c1=b2=c3=b4= ...,
      । ᫥⥫쭮 A ,  襭 㪢
-  b1  c1.
     ਬ: B='abab',  C='baba'.   ⢥ A      
'babab', ⠪  'ababa').
     ᫨  (*)  믮,     A,  ભ
㪢.    室  ⠪    B'  C',   c(i)=b(i+1) 
b(i)<>c(i+1) ( c(i)<>b(i+1),  b(i)=c(i+1)),  砥 砫-
  A=b(1) ... b(i)b(i+1)c(i+1)b(i+2)
... b(k) ( A=c(1) ... c(i)c(i+1)b(i+1)c(i+2) ... c(k)).
     ਬ: B'='bc'
             C'='cd'
       ண 砥  B'  C' - 'bcd'.

                             16.
       ᫥⥫쭮    ⠪:
                2, 4, 8, 16, 32, 64, 128, ...
     । 童 ᫥⥫쭮 - 28.

                             17.
      ᫥⥫쭮   ᫠ 6  筮, -
筮, ⢥筮,  筮, 筮, ... ⥬ ᫥-
.  ⠫ 童 ᫥⥫쭮 ࠢ 6.

                             18.
     ᫨   ﭨ R  ன 窨    窨  ,
  窠  A  㤥  -  㦭 O(R,C) ࠤ R 
業஬  窠 C.
     쬥     窨 - V  W. 
     1. ﭨ R1 -  窨 V  窨 A.
     1. ﭨ R2 -  窨 W  窨 A.
     1. ﭨ R3 -  窨 V  窨 B.
     1. ﭨ R4 -  窨 W  窨 B.
     窠 A -        祪  祭  㦭⥩
O(R1,V)    O(R2,W).  窠 B -     祪 祭
㦭⥩ O(R3,V)  O(R4,W).  稬 窨 祭 -
⥩ ᮮ⢥⢥  १  A1,  A2   B1,  B2.   窨 
ᬠਢ  設  ࠢ७  ࠯樨    ᭮-
-१ [A1, A2]  [B1, B2]. (࠯ ࠢ७ ⠪ 
窨 A1  B1 ᨬ ⭮⥫쭮   窠 A2   B2
ᮮ⢥⢥). १  [A,B], 䨣騨  , -  
᫥ :  [A1,B1], [A1,B2], [A2,B1], [A2, B2] - .. 
  ஭ ࠯樨,   .  ⠪   ࠢ-
७ ࠯樨 । ௥   ஭ 
। ௥    ᥪ   -
 ( !),   宦 ⮩ ᪮ 窨 -
  ᮢ.

                             19.
     । 砫  樨,        ।
䠫訢    訢.   ᤥ, ᫨:
             ⠭⭠ ( 
䠫訢);
            ⥭  䠫訢 
(.. ଠ  ⮬,     ).
                             :
     쥬 13   3 : 4,4  5 .
     1.      ࠢ.   ⠭ -
. 㤥   १    䠫訢  -
१ .
      2- 訢
             (⠫ ।묨 ⮫쪮   쥩
         ).
     ᫨   ࠢ,    䠫訢 । ⠢ .  ᫨
,  । 襭 ,    ⥭.
     2.            ࠢ.    
쥩  - ⠭ .  ।  襭  
   (殮)   ()  ᨬ  १⮢ -
訢.
      2- 訢
           
᫨  ࠢ,   䠫訢 । ⠢  .  ᫨
,   ᫥騥 樨.
     2.1.   . ⭮,  䠫訢  室
।  殮 .
     2.2.  殮 .  ⭮,   䠫訢  -
 ।  : 殮  ᫥   ࠢ.
     ⮬ 䠫訢    ।   ⠢襥
訢.

                           20.
{                      "모    ஢"
      짮⥫ 뢠 ᫮  4 ,     
1  6,  祬   ࠧ.  ࠡ  ,  
㣠뢠  ᫮  ᫥騬 ࠢ:  뢮 ᫮  짮-
⥫ ᮮ頥,  ᪮쪮   "몮"  "஢",  ..  ᪮쪮 
   ᢮   ᪮쪮  ᮤঠ   ᫠,  
ᮢ   祭.  ਬ,     ᫮  1264,
襭 1256.  ⮬ 砥 2 몠 (1,2)   ஢ (6)
     ) ࠢ   ,  뢠 6-筮 ᫮  -
  1  9, 祬 ।  ᪠ .

     ਬ砭 : 襭 ᫮  㤮⢮ ࠢ 
뢠;   室  1  .

     ⠪, 1-  室     ᫥饬  ࠢ:
뢠 ࢮ 襥 ᫮, ஥   㬠, ..
 ᮯ⠢  ࠭ 襭 ᫠     -
 ⠪- ⢮ '몮'  '஢'. ᫮ 㤥 ।⠢-
   ᨢ . }
 const  MaxSgn = 6; {  ᨬ쭮 祭  ᫥ }
          Sgn = 4; { ⢮   ᫥ }
type S = 1..MaxSgn;
     Numb = array[1..Sgn] of S;
function cows(n1,n2:Numb):byte;
 { ࠢ  ᫠  頥 १ ࠢ  
   <⢮ 몮>*10+<⢮ ஢> }
var i1,i2 : 1..Sgn;
    a : byte;
begin
 { 室 ࠢ   ࢮ ᫠  ᥬ ࠬ ண }
 a:=0;
 for i1:=1 to Sgn do
  for i2:=1 to Sgn do
   if n1[i1]=n2[i2] then
     if i1=i2 then a:=a+10 { ⨫ '' }
     else inc(a);
 cows:=a;
end;
type Step = Record  { ଠ    - ⢥ }
       n : Numb;  { 襭 ᫮ }
       answer : byte;  { ⢥ }
     end;
     Game = array[1..32] of step;
     { ଠ    - ⢥ }
var Nstep : byte; {  ⥪饣 蠣 }
    Info : Game;
    Curnumb : Numb;
    i,j : integer;
    b : boolean;
BEGIN
 clrscr;
 for i:=1 to Sgn do Curnumb[i]:=i;
 Nstep:=0;
 while true do
  {    ॡ࠭  ᫠    襭 }
  begin
   {  ⥪饥 ᫮ }
   for i:=1 to Sgn do write(Curnumb[i]);
   write(' ? ');
   inc(Nstep);
   Info[Nstep].n:=Curnumb;
   readln(Info[Nstep].Answer);
   { 짮⥫   ᫮, ࢠ 
     ண '모',  - '஢' }

   if (Info[Nstep].Answer div 10+Info[Nstep].Answer mod 10)>Sgn then
    begin writeln('  !'); exit; end;
   { '몮'  '஢'     , 祬  }
   if Info[Nstep].Answer=10*Sgn then exit;
   {     ᫠ }
   repeat
    i:=Sgn;
    { 祭 ᫠  1- }
    while (i>=1) and (Curnumb[i]=MaxSgn) do
      begin
       Curnumb[i]:=1;
       dec(i);
      end;
    if i<1 then
      begin {  ᫠ ॡ࠭,  ⢥   }
       writeln(' ࠢ쭮 ⢥砫 !'); exit; end;
    inc(Curnumb[i]);
    b:=true;
    { ஢ઠ  騥   }
    for i:=1 to Sgn do
     for j:=i+1 to Sgn do
      b:=b and (Curnumb[i]<>Curnumb[j]);
    for i:=1 to Nstep do b:=b and (cows(Curnumb,Info[i].n)=Info[i].Answer);
    until b;
  end; { while }
END. { program }

     ⠪,   ணࠬ  ࠡ⠥  ᫥騬  ࠧ:    ⥬
᫥⥫쭮 㢥祭  1 㥬   6-
᫠,  ⡨ࠥ ।  ,     ࠧ, , -
,     ,   㤮⢮ ࠭騬  ᨢ Info
⢥⠬,  짮⥫  ⢥ .   -
 ᮢ: ᪮쪮 ᥣ   4-   -
    ᮤন ७.
      ᪮쪮  蠣   㣠 ⢥ ᠬ   
᪮쪮   ⥣?
      ⠥  ⢥  .  ⠪ ᪮쪮 ᥣ ᥫ?
      k   m 権.   ࢮ 樨    
  k  ,      k ਠ⮢.   ன-⠪   k
, .. k^2.  ⠪  m ࠧ, .. ᥣ k^m ਠ⮢. 騬
 .
     ।: ࠧ饭  n ७  k ⮢  m -
뢠 m- ᨢ ࠫ ᥫ,    
ॢ室 k.
     ⢥ত: ᫮  ࠧ饭    ७ﬨ  k  m ࠢ
k^m. ⥫⢮ ஢  樨:
      樨:  m=1   ஢ k ਠ⮢.
     ⨢ 室:   ⢥ত ୮  m=n-1.  -
,      ୮  m=n.  䨪㥬 ᫮ 1.  ࠢ  ⮬
 ਯ襬 k^(n=1) ࠧ饭  k  n-1. 筮 㯨 
1,2,3...k. 稬 k^(n-1)*k=k^n ਠ⮢.
      ࠧ,  ᫮ 4- ᥫ  ࠬ  1  6 ࠢ
6^4=1296.  ᬮਬ, ᪮쪮    ᮤন 
.
     ।: ࠧ饭   k ⮢  m 뢠 m--
 ᨢ   ண  ॢ室 k    ।
   . 祢,  ⢮ ࠧ饭 
   m<=k.
     ᫮ ࠧ饭  k  m ਭ  A.
     ⢥ত A=k*(k-1)*...(k-m+1)=k!/(k-m+1)!
     ⥫⢮ ஢  樨:
      樨:  m=1   ஢ k ਠ⮢.
     ⨢ 室:  ⢥ত ୮  m=n-1. 롥६
  k!/(k-n+1)! ࠧ饭  k  n-1. ᥫ,    
-(k-n+1). ਯ뢠  । ࠢ  ࠭ ࠧ-
饭  稬 (k-n+1) ਠ⮢. ॡ  ࠧ饭:
   (k!/(k-n+1)!)*(k-n+1)=k!/(k-n)!
      ࠧ,᫮ 4- ᥫ  ࠬ   1    6  
७ ࠢ A46=6*5*4*3=360,  ..   3 ࠧ ,  祬 ᫮
ਠ⮢,    ॡࠫ. ⠪  諨  ᯮᮡ  -
⨬樨 襩 ணࠬ: ஢   ᫠,   , -
  ᮤঠ  . 쬥   ,  ᥩ-
 ஡㥬 業 ᨬ쭮 ᫮ 蠣,  ஥ ⣠뢠-
 訩 ப. ਠ⮢ ⢥  :
      㣠  4  .  ।      2,1,0 "몮".
 㣠 3 .  ।    3,2,1,0 "몮".   ⠪
: 砥 3+4+2+1=10 ਠ⮢.
      ࠧ       ⢮  ⨬  ᥫ
㬥蠥,  ᫨  ᬠਢ 訩 砩,   祬  10
ࠧ.  ᫮ 蠣,   ஥ 㣠뢠 訩 ப,    祬
[log10 360]+1=3
       ⥯ ⠥  䥪⨢ ࠡ ணࠬ.
 㦥 뫮 ⬥祭 ,   筮 ॡ  1096 -
権,   ᥣ  360.    ࠧ    㣠뢠,
.. ᫥ 蠣, ⠪    ⥣ " 訩 
室",  㬥 ६ 뢠 室.
     ஢ ᫠ 㤥 ᫥騬 ࠧ:   砫 롥६
⢮ ,஥   室,    ⥬ 㤥 ⠢
 ⮣ ⢠  ᮡ. ⢮  㤮,  -
 楫,  ।⠢   ᨢ   4,    ண
ᯮ  浪 ⠭. 㤥 ஢  ᨢ 
ᨪ᪮ 浪, .. ᭠砫  ࠢ    -
,᫨  ࠢ -- ,  ⠪ .  :
     1   2   3   4
     1   2   3   5
     1   2   3   6
     1   2   4   5
     1   2   4   6
     1   2   5   6
     1   3   4   5
       .   .   .
       .   .   .

,   㢥 ᠬ ࠢ ,    ,
 㬥 ⥬  ,     .  ⨢,  
,   㦭 㢥稢,  㬥蠥  1,     ⮫쪮
᫥   ᨢ  㤥 㢥, ᪠窮ࠧ -
 ࠢ  ᨢ, 室  ᫥饬 :
    {n-᫮ }
    {A-ᨢ  ᮤন ⥪騥 樨}
   Var
      p:integer;
        {  , 㢥稢    }
   Var
      i:integer;

     for i:=1 to n do
       a[i]:=i;{ 砫쭮 祭 }

     p:=n;{ 㢥稢 ࠩ ࠢ  }

     while p>=1 do
      begin
         {Write(A)}-{᫥⥫쭮 ⮢};
         for i:=k  downto p do
           A[i]:=A[p]+i-p+1;
         if a[k]=n then dec(p);
                   else p:=k;
      end;

 ஢ ᥢ ⠭ ⢠ 㦨 ᫥-
 :     ନ஢ ᥢ ⠭  
N-1  .⠢  ⠢訩    ⠭   -
 樨,  稬  樨  n. ਬ, ᫮ 1
 ⠭ 2 3  ⠢ ᫥騬 3- ᯮᮡ:
     1  2  3
     2  1  3
     2  3  1
 ⬥,    騥  ⠭  
᢮⢮:  ⠭ 砥  ।饩 ⥬ 
 ᥤ ⮢.  ਬ   ४ᨢ,   稬,
ਬ, ᫥ ᫥⥫쭮 ⠭:
     1  2  3
     2  1  3
     2  3  1
     3  2  1
     3  1  2
     1  3  2
 ਬ  ⮣  ⬠  ४ᨢ ଥ  
࠭ ᥢ ⠭.   ⠥ ਢ 
    ४ᨢ  ଥ.   i-   室
:

    㤠  ᥩ  - ࠢ  .

        n-i+1   ⠭    n-i,
    ஥  ᥩ .

 室  ᫥饬 :
  { B - ᨢ ⮢ ⠭ }
  { Direc - ᨢ ࠢ, i-  ࠢ TRUE, ᫨ ⥫ i 
    ࠢ }
  { Posit - ᨢ  i-   ⠭ n-i ⮢ }

    for i:=1 to n do
      begin
        B[i]:=i;
        Posit[i]:=1;
        Direc[i]:=TRUE;
      end;
    Posit[n]:=0;
      ⠥ ਢ    ४ᨢ  -
.  i-   室 :
     㤠  ᥩ -ࠢ  .
        n-i+1   ⠭  n-i,  -
஥  ᥩ .
   室  ᫥饬 :
  {-ᨢ ⮢ ⠭}
  {Direc-ᨢ ࠢ. i-  ࠢ true,
   ᫨ ᫮ i  ࠢ }
  {Posit- ᨢ  i-   ⠭
   n-i ⮢}
For i:=1 to n do
 begin
  b[i]:=i;
  posit[i]:=1;
  direc[i]:=true;
 end;
posit[n]:=0;
i:=1;
while i<n do
 begin
  direct[i]:=not direct[i];
  posit[i]:=1;
  if direct[i] then inc(x);
  inc(i)
 end;
if i<n then
 begin
  if pirof[i] then k:=c[i]+x
              else k:=n-i+1-c[i]+x;
              b[k]<->b[k+1];
              inc(posit[i])
 end
end;
  , ,  ਠ ணࠬ 
Const MaxSgn=6;
         Sgn=4;
Type s=1..MaxSgn; {}
     Numb=array[1..Sgn] of s;
function cows(n1,n2:Numb):byte;
 { ࠢ  ᫠  頥 १ ࠢ  
   <⢮ 몮>*10+<⢮ ஢> }
var i1,i2 : 1..Sgn;
    a : byte;
begin
 { 室 ࠢ   ࢮ ᫠  ᥬ ࠬ
   ண }
 a:=0;
 for i1:=1 to Sgn do
  for i2:=1 to Sgn do
   if n1[i1]=n2[i2] then
     if i1=i2 then a:=a+10 { ⨫ '' }
     else inc(a);
 cows:=a;
end;
type Step = Record  { ଠ    - ⢥ }
       n : Numb;  { 襭 ᫮ }
       answer : byte;  { ⢥ }
     end;
     Game = array[1..32] of step;
     { ଠ    - ⢥ }
var NStep:byte;
    Info:game;
Procedure testnumber;
{楤 ஢, 㤮⢮  室 ᫮ ࠭ ⢥⠬,
 ,  ᫨ ,     ⢥ .  砥, ᫨ ᫮
 㣠, 稢 }
Var i:byte;
begin
 for i:=1 to nstep do
  if cows(n,info[i].n)<>info[i].answer then Exit;
 inc(Nstep);
 info[nstep].n:=n;
 for i:=1 to sgn do write(n[i]);
 write('   ?   ');
 readln(info[nstep].answer);
 if info[nstep].answer=sgn*10 then
                              halt;
                              { 祭}
 end; {test number}
Var tset{⥪饥 n-⭮ ⢮}
    ,tn:number;
    i,j,k,l:byte;
    direc:array[1..sgn] of boolean;
    posin:array[1..sgn] of byte;
begin
{樠}
nstep:=0;
for i:=1 to sgn do tset[i]:=i;
i:=sgn;
while i>=1 do
begin
 tn:=tset;
 for j:=1 to sgn do begin
 posit[i]:=1;
 direct[i]:=true;
 end;
posit[sgn]:=0;
j:=1;
testnumber(tn);
while j<sgn do
 begin
  j:=1;
  k:=0;
  while posit[j]=sgn-j+1 do
   begin
    posit[j]:=1;
    direct[j]:=not direct[j];
    if direct[j] then inc(x);
    inc(j);
   end;
  if j<sgn then
   begin
   if direct[j] then l:=posit[j]+n;
                else l:=sgn-j+1+posit[j]+k;
   inc[posit[j];
  end;
 end;
writeln(' ');
end.

                             21.
       ஢  ⮫쪮 ࠡ ணࠬ  ४-
 ,    ४. ਬ:
     a    b    c
     0    0    0           㣮쭨!
    -1    1   0.5
     1    1    3
     0    1    2
         ।?

                     22 (஢ ..).
     稬 ᫮ १ F(x,y):
      x=2, y=4.  F(2,4)=True,  F(6,-2)=False.
      x=6, y=-2. F(6,-2)=True, F(4,8)=False.
.. F(6,-2)  ࠢ  True  False ६!  -
⥫쭮 ⠪ < ᫮ 믮 樨 >  . -
 ⬨᪨  ࠧ訬.  ᫨  ࠧ訬 ᯮ짮
 ᫮ 饭  㭪樨 F(x,y),    頥  ᥣ
᪮  祭  False,    㬥 x  y ॡ㥬
ࠧ,  ⥫ 横  㤥 믮,   ⮬ 砥 -
  .

                          23.
     ⮤ 1.    㧫 (㡪) 奬      
ﭨ Gi'.  ࢮ砫쭮 ⠭ G1,  G2  G3  G1',  G2' 
G3',  뢠,   室,  ਪ १ 室 A,  B  
ᮮ⢥⢥.   ⮣, ⮡ G1,G2  G3 ⠢  ﭨ
G1', G2'  G3' 室, ⮡ १   室 뢠-
  ⭮ ᫮ ਪ.
     ᫨ ᫥  ਪ,  襭묨  ஢ ⮬,
   ,  १ 室, ਬ, ABC 祭
१ 室 BCA,  ..   浪 ᪨ ਪ 祣  
!
     ᬮਬ 㧫,ﭨ     -
 ਪ १ 室 A, ..  ࠭ ⮤ 뢠-
 ⮫쪮 ⭮ ᫠
               A

               *
               |  \
               |    \
               |      \
               |        \
               |          \
               |            \
               |              \
               |                \
               |                  \
               |                    \   G4
               |                     /*
               |                   /    \
               |                 /        \
               |               /            \
               |             /                \
               |           /                    \
               |         /                        \
               |       /                            \
               |     /                                \
               |   /                                    \
          G6   | /                                        \  G7
               *                                            *


ਪ १ 室 ᥣ G1=G1',   ﭨ G6  G4  G7  
  2^3=8,   000  111. ᫨  㤥 뢠 १ A
 ਪ,     ਨ ᨬ  8  ਪ  -
- ﭨ G6 G4 G7  易⥫쭮   -
 ਨ,    ﭨ   横᪨  ,  
砩  뢠  8  ਪ ᢮    -
⢮ ,  ॢ室騬 8.
     筮,   뢠 ਪ १ B  砥 -
    祬 2^5=32  ਪ,   殮 ன -
 ⠭   ﭨ G4 G5 G6 G7 G8,   -
 C ,    A, 㤥     8 .
     ॡ     樨 A2x B2y C2z 뢠
ਪ ( A2x 砥   2X 뢠 ਪ १
室 A,  0<= x,y <=8,  0<=z<=32 ) 砥  -襭
,  ।塞 ⢨ 襭.
     ⮤ 2.   ᬠਢ      .
設    Y-㡪,   㣨 - ᮥ 
㡮. ⠪,      8 設,    -
  ਯᠭ   ⮪ 0   1.  ᥣ    2^8=256
ᥢ ﭨ ࠫ쭮 ⮬.  ﭨ 
।⠢   8- ⭮ ᫠.      256
設 ᫥騬 ࠧ:
    1) ⨬ 設 ࠬ  0  255.   設 
 ᮮ⢥    ﭨ ⥬ Y-㡮.
    2)   ﭨ ⥬ i -  0  255 
        1.  ﭨ i  ਪ १ 室 . ⥬ -
        ३  ﭨ iA.    ᮥ ਥ஢
        㣮 設 i  iA.    ᨬ .
        2.  ﭨ i  ਪ १ 室 B. ⥬ -
        ३  ﭨ iB.    ᮥ ਥ஢
        㣮 設 i  iB.    ᨬ B.
        3.  ﭨ i  ਪ १ 室 C. ⥬ -
        ३  ﭨ iC.    ᮥ ਥ஢
        㣮 設 i  iC.    ᨬ C.
    3)  設  ,  ᮮ⢥饩 砫쭮  ﭨ,
 砩訩  (  )  設, ᮮ⢥-
 筮 ﭨ.  ⪨   ⮬   -
  १ 室 ਪ. ᫨   ,  -
 襭  .
      ᪠ 砩 ⥩ .    "  -
".

                              24.
      襭    ⠪ 뢠 㭪 ⪠-
 F(j),  ਭ饩 楫᫥ 祭 [  .  . -
஥    ᫨⥫  ⬮.,  .,  1979,  .386].
㭪 F  ⠪: F(j) -- 襥 ᫮ s<j,  ண
A1 A2 ... As -- 䨪 楯窨 A1, A2, ..., Aj, ..
        A  A  ... A  = A      A      ... A
         1  2      s    j-s+1  j-s+2      j

  ᫨ ⠪ s=>1 ,  F(j)=0.

                         (m)
   । 㭪    F   (j) ᫥騬 ࠧ:

           (1)
       1) F   (j)=F(j)

           (m)       (m-1)
       2) F   (j)=F(F     (j))  m>1.

   㤥   㭪  F ⨢.   । F(1)=0.
⨬,  ᫥ F(1), F(2), ... , F(j).  F(j)=i. -
 ᫨ F(j+1) ᫥㥬 A     A    . ᫨ A    = A    , 
                               j+1     i+1        j+1    i+1
F(j+1)=F(i)+1 ,

   ᪮   A  A ... A  A   = A      A      ... A  A   .
                1   2    i  i+1   j-i+1  j-i+2      j  j+1

   ᫨ A   <> A   ,  ⮣   襥 祭 i, 
         j+1    i+1
ண
              A  ... A  = A      ... A    A    = A   ,
               1      i    j-i+1      j     i+1    j+1
᫨ ⠪ i .  ᫨ ⠪  i  ,    祢,  
F(j)=0.  i , i , ... -- 襥, ஥  稭  ..
               1   2
祭 i,  
                 A  A  ... A  = A      ... A .
                  1  2      i    j-i+1      j

       ⮩ 樨 㡥, 

                      (2)                       (k)
   i =F(j), i =F(i )=F   (j),  ..., i =F(i   )=F   (j), ....
    1        2    1                  k    k-1


⠪,  砥 A   <> A    室 襥 m,  ண 
                j+1    i+1
          (m)
      1) F   (j)=u    A    = A   ,   ⮣ F(j+1)=u+1,
                        j+1    u+1

          (m)
      2) F   (j)=0    A    <> A ,  ⮣ F(j+1)  ࠢ 0.
                        j+1     1

      ᫥ F ᫥騩:

         begin
            F[1]:=0;
            for j:=2 to n do
               begin
                  i:=F[j-1];
                  while ( A[j]<>A[j+1] ) and ( i>0 ) do
                     i:=F[i];
                  if ( A[j]<>A[i+1] ) and ( i=0 )
                  then F[j]:=0
                  else F[j]:=i+1;
               end;
           end.

        F  O(n) 蠣.  ⮡  ,
,  ᫮  while  ࠡ ணࠬ ( -
樮쭠  㬥襭 祭 i ஬ i:=F[i] ) -
   N.
   ⢥ ᯮᮡ  㢥 i  ⮬ ࠣ ணࠬ --
 믮  F[j]:=i+1    쥩      ணࠬ
ப.     믮   N-1 ࠧ, ᫥-
⥫쭮,   㬥襭 i   while  ந  -
 N ࠧ.  ⠫쭠  ⬠  ᫮ O(n),  -
⮬     ᫮ O(n).
    ⠪,  ᫨ F(n)=U,  砥, 
              A  A  ... A   = A     ... A                      (*)
               1  2       U    N-U+1     N
     ⮣, ⮡ ᫥⥫쭮
              A  ... A
               1      N
뫠 ਮ᪮ 室   筮,  ⮡    饩
  ப  (*)
                         A      ... A
                          N-U+1      U
뫠 ⭠  ਮ ( ⢥ ਮ ᫥⥫쭮
 㯠 ப A  ... A   ).
                      1      N-U

    ⢨⥫쭮, ᫨ A  ... A   = A      ... A      ,  
                         1      N-U    N-U+1     2(N-U)

᢮ (*)   A      ... A       = A         ... A         ..
                 N-U+1      2(N-U)    2(N-U)+1      3(N-U)

    ⥫쭮 襭 

      1)   l ४뢠饩  ப  (*)
            l=U-(N-U+1)+1 =2*U-n;
 砥 l<=0 ᫥⥫쭮 A ... A  ਮ᪠. ⮯.
                                  1     N
      2) ᫨ l>0 ,  ஢ਬ,    楫 l   -
ਮ (..  N-U)
     l mod (N-U) = (2U-N) mod (N-U) = - ((N-U)-U) mod (N-U) =
                         =U mod (N-U) = R
   ᫨ R=0,  ᫥⥫쭮 ਮ᪠  ਮ
                           A ... A   ,
                            1     N-U
 -- .


                           25.
      ᪮筮 ࠦ , ᫨  ।⠢-
   (S),   S - ࠢ쭮 ᪮筮 ࠦ.  祢,
  ⮥ ᪮筮 ࠦ  ⢥ ࠧ
।⠢  
                               ES,

     E - ୮ ᪮筮 ࠦ;
     S - ᪮筮 ࠦ (, ⮥).
     稬 १ S(n) ᫮ ࠢ ᪮  ࠦ  
2n  ᪮.    ᫮   ᪮ ࠦ  2n
᪮  S(n-1).  ࠧ
            n
      S(n)=SUM(S(i-1)*S(n-i)).
           i=1
     ,  S(0)=1,  -  ⮥ ᪮筮  ࠦ,
 室   ᫮ O(n^2).

       ⮩ :
     㤥 ࠦ  / 뢠 ᪮,  \ - 뢠-
.  ࠢ쭮 ᪮筮 ࠦ ()  n=1 㤥 룫拉
⠪:
                                /\
                               0  1
     ᫨  쬥 ⨭
                               / \
                             / \ / \
                            0   1   2

    n=2  ⠪:

                       / \ / \         /\
                       0   1   2       /  \
                                       0   2
-     ᮮ⢥ ᢮ ⢥  -
    設 0  2.  筮,     n=3 
뢠 ﬨ  
                               / \
                             / \ / \
                           / \ / \ / \
                          0   1   2   3

 .. ⠥ ᫮  (᫮ ⥩)  n=3.
      C ->    1            設 A    ⢥-
            /   \          ᯮᮡ -  窨 0, ࠢ  B,C 
    B ->   1     3  <- D   1  , ᮮ⢥⢥,
          / \   / \       ⮫쪮  祪 A, B, A.   E 
  A ->   1    2    5         1, ⠪    B, ⮬
        / \ / ^ \ / \     ⪠ E  㬬 ⮪  㧫,
       1   1  E  2   5    .. 2.  D     C
       0   1     2   3    (⪠ 1),   E (⪠ 2), ,
                          ᫥⥫쭮, ⪠ D=3  ..
     筮 室 ⢮    n.

                           26.
        ⭠  s ஢ ⥪,    -
室  ࠧ  s  ᥢ  ⥫ ( ਬ,
 s=6    1*6,  2*3, 3*2, 6*1 ), ९ ஢-
  ⥪    ⠡  T  ᮮ⢥饣  ⥫ ࠧ 
⮫栬  ᬮ,     祭 ⥪ ( ᫨  -
  ப ),   ᫮  .
     室 ஢    ⥫, ⠪ 
⥪   ஢ 筮.    ।ᬮ
ࠡ ,  ⥪   .
     ᫨ s ᫨誮 讥,   ࠧ [1..s, 1..s] 
    設.   ⮬ 砥   ஬-
    ⥪  s   ᨢ T[1..n,  1..m],  nm=s;
 ⮬ 饭   [i,j] ⭮ 饭  -
 (i-1)*n+j ஢ ⥪(।⠢ ᥡ,   
⥪ ࠧ  ᪨  m - ப   ,  j-  
i- ப  ᨢ T 㤥 ᮮ⢥⢮ 㪢 (i-1)*n+j -
஢ ⥪.

                           27.
       ⬮ 襭  ⠪:
    1. ।,    ⥪, ஢ ண -
筠,  ᫥⥫쭮,    ⥪ 쭮 ,
 ண   襩   ࠧ
             b[1] b[2]...b[k] = c[1] c[2]...c[m]
 ᫮ ⢠ A.  ।⠢ ⥪    १  DE,  
ࠧ   ࠧ ᫥騬 ࠧ

         b[1]   b[2]               ...                  b[k]
    D  Ĵ  E


          c[1]    c[2]  c[3]        ...                  c[m]
    D  Ĵ  E


      ⮣,  ⥪ 쭮 , ᫥,   ᫮
 ࠧ ࠧ      ⨪ (஬ 
⥪),  ..  b[1]  <>  c[1],      ᫮ (ਬ
b[1])  室  㣮  ⢥ 䨪  ।⠢
    c[1] = b[1] p[1] ,   p[1] -  ⠢  ᫮
(䨪).  ,  p[1] 室  b[2] (b[2] =p[1] p[2]), 
b[2] 室  p[1] (p[1] = b[2] p[1] )  ⢥ 䨪.  -
塞  䨪 p[2]. த 뤥 䨪  䨪,
砥,      - 蠣 p[j] ᮢ    
᫮.
     㦤  ᭮ ᫥饣 ⬠ :
     㫥 蠣 쬥   (a[i], a[j]),  i <> j, ⠪
 ᫮,       䨪 㣮,    
䨪 p[0,k].
     j- 蠣    (p[j-1,k],  a[i]),     ᫮
 䨪ᮬ 㣮,   室  䨪,     
,    﫨  । 蠣 ⬠, -
稬 p[j,k].
     蠣 塞    ,  - 䨪 
ᮢ     ᫮ ( ⮣   筮
㥬  ⥪),      । 蠣   
  䨪  (  ⮣      ⥪  
⢥ ஢).
    .. ⢮  ᫮ ࠭祭,    䨪ᮢ ⠪
筮  ᫮,     - 蠣  易⥫쭮 ⠭-
.
      ਭ  ࢮ砫쭮 ਠ A.  न-
  . ᮭ (. ୥᪨ ᡮ୨, .3 . 93-
102 ).
    2. ᫨  ⥪,  ,  ᯮ A,  
筮,   稭 뢠  A ᫮   樨,
  ⢮ A'  ᨬ  ᫮  ᫮  ⠪,
  ⥪, ⠢  ᫮ A'  筮.
    砫  A 㤠塞 i- ᫮ ,  i = 1, 2, ... , n,  
஢  㭪 1. ᫨ 㤠  ᫮  A   -
砥 ᪮ ⢠ A',   ⮣ 㥬  ⠭  
n-2 ᫮  ⢠ A,     祭 ⢠ -
 ஢  㭪 1,  ..
    ᯮ㥬 ᫥騬  ⬮  樨  ⠭  i
᫮  ⢠ A:
     ᨢ  B    室  ᯮ㥬  
蠣 ⮢  A (饥  ᫮ i).   ⢥ 砫쭮 -
䨣樥 쬥 ᫥: B[j]=j, j=1,...,n.
    饬 B[j]  ᨬ ᮬ j ⠪, B[j]<n+j-i, 㢥-
稢    B[j]    1,     k>j  B[k]=B[k-1]+1
(B[j]   ⮬ j,   饬  㢥稢  1 ⠪ B[j] 
ᨬ ஬ j,  ⮡   騬 -
ﬨ ⮢ ᨢ B[k],  k>j, ᫥  B[i]  -
室  n). ᫨ ⠪ B[j]  ,   -
⠭   i 祭.
    稭     i=n-1   ஢   㬥襭 i 
 ,    i  ⠭  ࠢ  1  (⮣  A'  ⮨  
⢥  ᫮),    ஢ઠ  㭪 1   -
筮 ஢.
    3.  㭪 ॠ ᫥騬 ४ᨢ ⬮ -
᪠  饭 ( ᫮ ᨢ A 㯮冷祭   ࠬ):
    ) ᫨ ப ,      ஢ -
,   ࠧ ⥪  ஢塞 a[i] (  i 
1  n)  宦  砫 㥬    ப.
    ) ᫨ - a[i] 室  ப  䨪,   -
  i ⮣ ᫮,  ⥬ 뤥塞 ᫮    ப,    
⪮ ⥪ ந  ࠧ  㭪 ).
    ᫨    [i]  室  ⢥ 䨪  -
 ᥩ ப,   ⢫塞   㭪 ), ।-
⥫쭮    砫 ப ᫥ 㤠 㤠 ᫮,
 ⠥ 뤥  ⥪ ᫮  訬 ஬    A.  ᫨
 ⢨  (. .  室  砫 室-
 ப),    稢 ᢮ ࠡ.    
஢ .
     1.  ᫥ ⥮६ (મ . .):
     筮 ஢ ⥪ 筮 믮 -
   ᫥ ᫮:
    1)             ᫮ (a[i], a[j]),
i<>j, ⠪,     ᫮  䨪 㣮.
    2)             ᫮ (a[i], a[j]),
i<>j, ⠪,      䨪 㣮.
     2.  ਢ ਬ ⠪ ⢠  ᫮, 
   믮  ᫮ 1,   ᫮ 2  ।饣
㭪,  ஢  ⥪, ⠢   -
 ᫮ - 筠.

                             28.
        ,    뫪 ᨣ  䥩
 ਥ ᨣ  ன ஥ ६.  ணࠬ -
室   ⮩ .

                             29.
     ⮡ 㬬 ⮪ ॡ  設  뫨  ࠢ,  室,
⮡    㬬 ࠢ﫠 2*(1+2+...+12)/8,  ⠪   ॡ
ᬥ 2 設 (..  뢠 ),  ⢮ -
設 - 8.  ᫮ 2*(1+2+...+12)/8=2*12*13/(2*8) -  楫. ⮬
㬥樨  .

                             31.
      ᠭ  襭  ⮩       ୠ
"ଠ⨪  ࠧ" N 5-6  1992 .

                             32.
    ନ㥬 ᨢ A   業 㯮  த. 祢, 
᪮  業    㭪⮢      ⮢
 ᨢ.
    1. ᬠਢ  ᨢ,     ⥪ 業
塞 㬬 ᤥ. 室 ᨬ  㬬 ᤥ.
    2. ᬠਢ  ᨢ,     ⥪ 業
塞 饥  ᫮  ⭨  ࣮.  室  ᨬ쭮
᫮ ⭨  ᮮ⢥ ⮬  ᨬ 業.

