Tabindah Model High School Reg.

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 Tower of Trans.    

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Where's the Math in this Game?

The number of separate transfers of single disks the priests must make to transfer the tower is 2 to the 64th minus 1, or 18,446,744,073,709,551,615 moves! If the priests worked day and night, making one move every second it would take slightly more than 580 billion years to accomplish the job! You have a great deal fewer disks than 64 here. Can you calculate the number of moves it will take you to move the disks from one of the three poles to another?

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

  • Only one disk must be moved at a time.

  • Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.

  • No disk may be placed on top of a smaller disk.

With three disks, the puzzle can be solved in seven moves.

Alternating between the smallest and the next-smallest disks, follow the steps for the appropriate case:

For an even number of disks:

  • make the legal move between pegs A and B
  • make the legal move between pegs A and C
  • make the legal move between pegs B and C
  • repeat until complete

For an odd number of disks:

  • make the legal move between pegs A and C
  • make the legal move between pegs A and B
  • make the legal move between pegs C and B
  • repeat until complete

In each case, a total of 2ⁿ-1 moves are made.

 

InterAction

Education

 
 Tic-tac-toe

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Tic-tac-toe (or Noughts and crosses, Xs and Os) is a paper-and-pencil game for two players, X and O, who take turns marking the spaces in a 33 grid. The player who succeeds in placing three respective marks in a horizontal, vertical, or diagonal row wins the game.

When considering only the state of the board, and after taking into account board symmetries (i.e. rotations and reflections), there are only 138 terminal board positions. Assuming that X makes the first move every time:

  • 91 unique positions are won by (X)
  • 44 unique positions are won by (O)
  • 3 unique positions are drawn

A player can play perfect Tic-tac-toe (win or draw) given they choose the first possible move from the following list.

  1. Win: If the player has two in a row, they can place a third to get three in a row.
  2. Block: If the [opponent] has two in a row, the player must play the third themself to block the opponent.
  3. Fork: Create an opportunity where the player has two threats to win (two non-blocked lines of 2).
  4. Blocking an opponent's fork:
    • Option 1: The player should create two in a row to force the opponent into defending, as long as it doesn't result in them creating a fork. For example, if "X" has a corner, "O" has the center, and "X" has the opposite corner as well, "O" must not play a corner in order to win. (Playing a corner in this scenario creates a fork for "X" to win.)
    • Option 2: If there is a configuration where the opponent can fork, the player should block that fork.
  5. Center: A player marks the center. (If it is the first move of the game, playing on a corner gives "O" more opportunities to make a mistake and may therefore be the better choice; however, it makes no difference between perfect players.)
  6. Opposite corner: If the opponent is in the corner, the player plays the opposite corner.
  7. Empty corner: The player plays in a corner square.
  8. Empty side: The player plays in a middle square on any of the 4 sides.
 

More games will be added soon!