Saturday, May 30, 2015

Tic-Tac-Toe as a Magic Square

Here's a game that looks like a math puzzle, but is really tic-tac-toe in disguise.
    In the math game, two players take turns picking a number from one to nine. The numbers can only be picked once each game. The first player who picks three numbers that add up to fifteen wins.
    A game board is usually drawn up like this:

    1   2   3   4   5   6   7   8   9
    The players take turns circling their number picks using different colored markers. The first player that can total fifteen with three of their picks wins. A sample game could go like this:
    Player 1 circles 5
    Player 2 circles 7
    Player 1 circles 6
    Player 2 circles 4 (to block 5+6+4)
    Player 1 circles 8
    Player 2 circles 2 (to block 8+5+2)
    Player 1 circles 1 and wins 8+6+1
This is a game that is isomorphic to tic-tac-toe, but on the surface looks completely different. Two players in turn say a number between one and nine. A particular number may not be repeated. The game is won by the player who has circled three numbers whose sum is 15.

Magicsquareexample.svg Plotting these numbers on a 3×3 magic square shows that the game exactly corresponds with tic-tac-toe, since three numbers will be arranged in a straight line if and only if they total 15.

You don't have to memorize every square on the magic square, although that isn't too hard to do. Remember that the diagonal is 4-5-6 and the other two corners are 2 and 8. Play your regular game of tic-tac-toe, but pick the numbers that match up with the square you would normally place your X or O.

Just for fun:
Can anyone come up with a name for this game? Put it in the comments.

Friday, May 29, 2015

How to Tell Which Fraction is Largest

The secret is to cross-multiply the two fractions. Cross-multiplication is a handy math skill to know. You can use it for a quite few different purposes. You are going to use it to find out which of two fractions is greater.

To cross-multiply two fractions:
  1. Multiply the numerator of the first fraction by the denominator of the second fraction and jot down the answer.
  2. Multiply the numerator of the second fraction by the denominator of the first fraction and jot down the answer.
For example, suppose you have these two fractions:
 3              4
---  and    ---
 7              9
When you cross-multiply, you get these two numbers: 3 X 9 = 27 and 4 X 7  = 28
Make sure you start with the numerator of the first fraction. To find out which of two fractions is larger, place the two numbers you get, in order, under the two fractions. The larger number is always under the larger fraction.
 3              4
---            ---
 7              9
27            28
so 4/9 is larger than 3/7 (or 4/9 > 3/7).
What do you do if you have more than 2 fractions?
Answer: You take them two at a time.
For example, we have the fractions 5/13, 7/17, and 8/19. Which is the largest?
Take the first two and cross multiply:
 5                                7
---             and           ---
13                              17
5 X 17 = 85               7 X 13 = 91
so 7/17 is larger than 5/13.
Now take the larger of these two, 7/17 and compare it with the third, 8/19.  
 7                                8
---             and           ---
17                              19
7 X 19=133              8 X 17=136
so 8/19 is larger than 7/17 is larger than 5/13.

How to Never Lose a Game of Tic-Tac-Toe

There are many places on the internet that show you how to never lose a game of Tic-Tac-Toe. One of my favorites is Quora. You can click on it for FULL directions, including pictures. I'll go over some highlights.
IF YOU GO FIRST...Avoid placing your first piece on an edge square, and keep it on the center or a corner square. Placing it on an edge square will give your opponent the advantage.

IF YOUR OPPONENT GOES FIRST...Unfortunately, if your opponent goes first and uses all the above techniques, there's no way that you can win. In fact, the only way you can win is if his/her first move is an edge piece.

After the first moves by both players, you can follow the list below. Do the first thing on the list that you are able to do.

Win: If you have two in a row, you can place a third to get three in a row.
Block: If your opponent has two in a row, you must play the third to block your opponent.
Fork: Create an opportunity where you have two threats to win (two non-blocked lines of 2).
Blocking an opponent's fork:
        Option 1: You should create two in a row to force your 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, you should block that fork.
Center:  Mark 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.)
Opposite corner: If the opponent is in the corner, the player plays the opposite corner.
Empty corner: The player plays in a corner square.
Empty side: The player plays in a middle square on any of the 4 sides.

The above list is the programming for the first computer tic-tac-toe game. In 1952, OXO (or Noughts and Crosses, the English name for tic-tac-toe) for the EDSAC computer became one of the first known video games. The computer player could play perfect games of tic-tac-toe against a human opponent.

Can anyone comment on what a nought is?

Thursday, May 28, 2015


This game is called Chomp. It is normally played with just a table of squares on a piece of paper between two players, but I find it easier to understand by thinking of a chocolate bar like this:


             My favorite Chomp playing board

To play Chomp the first player chooses a square on the board, and then takes away everything above and to the right of it (you are taking a bite out of the top right corner of the chocolate bar). The second player then does the same thing with another remaining square. This process keeps continuing until all that remains is the bottom left square. The bottom left square is poisoned and the player who has to take it loses.

To understand how to play the game, click here to go to UCLA and play against their computer. It can be proven mathematically that the first player can always win, but the proof doesn't show how to do it. See if you can win at UCLA and come back and tell us how you did it.

Warning: This game is not really a game, but rather a unsolved proof that mathematicians have been trying to solve for quite a while. It is HARD. Do not spend too much time on it.

Credits: The game was originally stated by Fred Schuh in 1952 as his "game of divisors". David Gale reinvented this game. His version used an m-by-n (any size rectangular) chocolate bar. The name Chomp was invented by Martin Gardner in Scientific American.