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I'm not too clear on your requirements, but if the following is allowed, it may be a simple to follow algorithm. Probably a bug in there, especially since it uses a recursive function. Also, it checks if a door leads to the room you came from, but I don't know how a three room circular path would handle, it may just keep looping forever in those three rooms. You may have to add a history to ensure no room is checked twice. But by reading your description that may not be allowed.

solved = FALSE

SearchRoom(rooms[0],NULL)    // Start at room 1 (or any room)
IF solved THEN
  // solvable
ELSE
  // unsolvable
ENDIF
END

// Recursive function, should run until it searches every room or finds the exit
FUNCTION SearchRoom: curRoom, prevRoom
  // Is this room the 'exit' room
  IF curRoom.IsExit() THEN
    solved = TRUE
    RETURN
  ENDIF

  // Deadend?
  IF curRoom.doors <= 1 THEN RETURN

  // Search each room the current room leads to
  FOREACH door IN curRoom
    // Skip the room we just came from
    IF door.id <> prevRoom.id THEN
      SearchRoom(door, curRoom)
    ENDIF
    IF solved THEN EXIT LOOP
  NEXT

  RETURN
ENDFUNCTION

[Edit] Added 'id' to previous check, and updated to make more object oriented.

I'm not too clear on your requirements, but if the following is allowed, it may be a simple to follow algorithm. Probably a bug in there, especially since it uses a recursive function. Also, it checks if a door leads to the room you came from, but I don't know how a three room circular path would handle, it may just keep looping forever in those three rooms. You may have to add a history to ensure no room is checked twice. But by reading your description that may not be allowed.

solved = FALSE

SearchRoom(rooms[0], rooms[0])    // Start at room 1 (or any room)
IF solved THEN
  // solvable
ELSE
  // unsolvable
ENDIF
END

// Recursive function, should run until it searches every room or finds the exit
FUNCTION SearchRoom: curRoom, prevRoom
  // Is this room the 'exit' room
  IF curRoom.IsExit() THEN
    solved = TRUE
    RETURN
  ENDIF

  // Deadend?  (skip starting room)
  IF (curRoom.id <> prevRoom.id) AND (curRoom.doors <= 1) THEN RETURN

  // Search each room the current room leads to
  FOREACH door IN curRoom
    // Skip the room we just came from
    IF door.id <> prevRoom.id THEN
      SearchRoom(door, curRoom)
    ENDIF
    IF solved THEN EXIT LOOP
  NEXT

  RETURN
ENDFUNCTION

[Edit] Added 'id' to previous check, and updated to make more object oriented.

I'm not too clear on your requirements, but if the following is allowed, it may be a simple to follow algorithm. Probably a bug in there, especially since it uses a recursive function. Also, it checks if a door leads to the room you came from, but I don't know how a three room circular path would handle, it may just keep looping forever in those three rooms. You may have to add a history to ensure no room is checked twice. But by reading your description that may not be allowed.

solved = FALSE

SearchRoom(rooms[0],NULL)    // Start at room 1 (or any room)
IF solved THEN
  // solvable
ELSE
  // unsolvable
ENDIF
END

// Recursive function, should run until it searches every room or finds the exit
FUNCTION SearchRoom: curRoom, prevRoom
  // Is this room the 'exit' room
  IF curRoom.IsExit() THEN
    solved = TRUE
    RETURN
  ENDIF

  // Deadend?
  IF curRoom.doors <= 1 THEN RETURN

  // Search each room the current room leads to
  FOREACH room IN curRoom
    // Skip the room we just came from
    IF room.id <> prevRoom.id THEN
      SearchRoom(room, curRoom)
    ENDIF
    IF solved THEN EXIT LOOP
  NEXT

  RETURN
ENDFUNCTION

[Edit] Added 'id' to previous check, and updated to make more object oriented.

I'm not too clear on your requirements, but if the following is allowed, it may be a simple to follow algorithm. Probably a bug in there, especially since it uses a recursive function. Also, it checks if a door leads to the room you came from, but I don't know how a three room circular path would handle, it may just keep looping forever in those three rooms. You may have to add a history to ensure no room is checked twice. But by reading your description that may not be allowed.

solved = FALSE

SearchRoom(rooms[0],NULL)    // Start at room 1 (or any room)
IF solved THEN
  // solvable
ELSE
  // unsolvable
ENDIF
END

// Recursive function, should run until it searches every room or finds the exit
FUNCTION SearchRoom: curRoom, prevRoom
  // Is this room the 'exit' room
  IF curRoom.IsExit() THEN
    solved = TRUE
    RETURN
  ENDIF

  // Deadend?
  IF curRoom.doors <= 1 THEN RETURN

  // Search each room the current room leads to
  FOREACH door IN curRoom
    // Skip the room we just came from
    IF door.id <> prevRoom.id THEN
      SearchRoom(door, curRoom)
    ENDIF
    IF solved THEN EXIT LOOP
  NEXT

  RETURN
ENDFUNCTION

[Edit] Added 'id' to previous check, and updated to make more object oriented.

I'm not too clear on your requirements, but if the following is allowed, it may be a simple to follow algorithm. Probably a bug in there, especially since it uses a recursive function. Also, it checks if a door leads to the room you came from, but I don't know how a three room circular path would handle, it may just keep looping forever in those three rooms. You may have to add a history to ensure no room is checked twice. But by reading your description that may not be allowed.

solved = FALSE

SearchRoom(rooms[0],NULL)    // Start at room 1 (or any room)
IF solved THEN
  // solvable
ELSE
  // unsolvable
ENDIF
END

// Recursive function, should run until it searches every room or finds the exit
FUNCTION SearchRoom: curRoom, prevRoom
  // Is this room the 'exit' room
  IF curRoom.IsExit() THEN
    solved = TRUE
    RETURN
  ENDIF

  // Deadend?
  IF curRoom.doors <= 1 THEN RETURN

  // Search each room the current room leads to
  FOREACH room IN curRoom
    // Skip the room we just came from
    IF room.id <> prevRoom.id THEN
      SearchRoom(room, curRoom)
    ENDIF
  NEXT

  RETURN
ENDFUNCTION

[Edit] Added 'id' to previous check, and updated to make more object oriented.

I'm not too clear on your requirements, but if the following is allowed, it may be a simple to follow algorithm. Probably a bug in there, especially since it uses a recursive function. Also, it checks if a door leads to the room you came from, but I don't know how a three room circular path would handle, it may just keep looping forever in those three rooms. You may have to add a history to ensure no room is checked twice. But by reading your description that may not be allowed.

solved = FALSE

SearchRoom(rooms[0],NULL)    // Start at room 1 (or any room)
IF solved THEN
  // solvable
ELSE
  // unsolvable
ENDIF
END

// Recursive function, should run until it searches every room or finds the exit
FUNCTION SearchRoom: curRoom, prevRoom
  // Is this room the 'exit' room
  IF curRoom.IsExit() THEN
    solved = TRUE
    RETURN
  ENDIF

  // Deadend?
  IF curRoom.doors <= 1 THEN RETURN

  // Search each room the current room leads to
  FOREACH room IN curRoom
    // Skip the room we just came from
    IF room.id <> prevRoom.id THEN
      SearchRoom(room, curRoom)
    ENDIF
    IF solved THEN EXIT LOOP
  NEXT

  RETURN
ENDFUNCTION

[Edit] Added 'id' to previous check, and updated to make more object oriented.