1.4.2 The Clocks

The Clocks
IOI'94 - Day 2

Consider nine clocks arranged in a 3x3 array thusly:

|-------|    |-------|    |-------|    
|       |    |       |    |   |   |    
|---O   |    |---O   |    |   O   |          
|       |    |       |    |       |           
|-------|    |-------|    |-------|    
    A            B            C

|-------|    |-------|    |-------|
|       |    |       |    |       |
|   O   |    |   O   |    |   O   |
|   |   |    |   |   |    |   |   |
|-------|    |-------|    |-------|
    D            E            F

|-------|    |-------|    |-------|
|       |    |       |    |       |
|   O   |    |   O---|    |   O   |
|   |   |    |       |    |   |   |
|-------|    |-------|    |-------|
    G            H            I

The goal is to find a minimal sequence of moves to return all the dials to 12 o'clock. Nine different ways to turn the dials on the clocks are supplied via a table below; each way is called a move. Select for each move a number 1 through 9 which will cause the dials of the affected clocks (see next table) to be turned 90 degrees clockwise.

Move	Affected clocks
1	ABDE
2	ABC
3	BCEF
4	ADG
5	BDEFH
6	CFI
7	DEGH
8	GHI
9	EFHI

Example

Each number represents a time accoring to following table:

9 9 12       9 12 12       9 12 12        12 12 12      12 12 12 
6 6 6  5 ->  9  9  9  8->  9  9  9  4 ->  12  9  9  9-> 12 12 12 
6 3 6        6  6  6       9  9  9        12  9  9      12 12 12 

[But this might or might not be the `correct' answer; see below.]

PROGRAM NAME: clocks

INPUT FORMAT

Lines 1-3:

Three lines of three space-separated numbers; each number represents the start time of one clock, 3, 6, 9, or 12. The ordering of the numbers corresponds to the first example above.

SAMPLE INPUT (file clocks.in)

9 9 12
6 6 6
6 3 6

OUTPUT FORMAT

A single line that contains a space separated list of the shortest sequence of moves (designated by numbers) which returns all the clocks to 12:00. If there is more than one solution, print the one which gives the lowest number when the moves are concatenated (e.g., 5 2 4 6 < 9 3 1 1).

SAMPLE OUTPUT (file clocks.out)

4 5 8 9

{
ID:makeeca1
PROG:clocks
LANG:PASCAL
}
const ch:array[1..9,1..9]of 0..3=
              ((3,3,3,3,3,2,3,2,0),
               (2,3,2,3,2,3,1,0,1),
               (3,3,3,2,3,3,0,2,3),
               (2,3,1,3,2,0,2,3,1),
               (2,3,2,3,1,3,2,3,2),
               (1,3,2,0,2,3,1,3,2),
               (3,2,0,3,3,2,3,3,3),
               (1,0,1,3,2,3,2,3,2),
               (0,2,3,2,3,3,3,3,3));
var a:array[1..9]of byte;
     ans:array[1..9]of byte;
     i,j,k:byte;
     s:string;
begin
assign(input,'clocks.in');
reset(input);
assign(output,'clocks.out');
rewrite(output);
   readln(a[1],a[2],a[3]);
   readln(a[4],a[5],a[6]);
   readln(a[7],a[8],a[9]);
   for i:=1 to 9 do
     a[i]:=a[i] div 3;
   for i:=1 to 9 do
     for k:=1 to 9 do
       inc(ans[k],ch[i,k]*(4-a[i]));
   for i:=1 to 9 do
     ans[i]:=ans[i] and 3;
   s:='';
   for i:=1 to 9 do
     for j:=1 to ans[i] do
       s:=s+chr(i+48)+' ';
   delete(s,length(s),1);
   writeln(s);
close(input);
close(output);
end.