My TicTacToe Hypothesis

I've long wondered about the relation of the size of a TicTacToe grid and the number of squares to the outcome of a game.  For example, a 2x2 grid will always be won by the first player:

The odd numbers are the first player's moves, and the even are the second player's moves.

Part one of my hypothesis is that an n to the nth grid will always be won by the first player.  So far, so good!

If we now try a 3x3 (n+1 to the nth+1) grid, we get the following result:

Many people know that in 3x3 TicTacToe game that either the first player will win or a tie will result, especially if the first player always starts from the center.  In the above example, we have a tie because neither player has any moves that will lead to a win. The red numbers are blocking moves.  The only way player two would lose is by making his or her next move in the lower left corner.


Now let's try a three-dimensional game.  If the grid is 3x3x3, then player one can always win with careful play:

Again the red numbers are blocking moves and the green numbers are the winning moves.  The second player has only two choices for the fourth move (8), as shown or where the first player made the winning move (9).  Neither would have prevented the first player's win.

OK, let's try a 4x4x4 game.  After several tries where the first player won, I found one where the second player won, even though the first player kept trying to take the initiative:

The two red X's are the only choices the first player has, either of which still allows the second player to make the winning move.  Can you find which move by the second player put the first player on the defensive?

And thus goes my hope of a grand theory of TicTacToe crashing to the ground, pulled down by the weight of contrary evidence.  My hypothesis is that I went too long to test my TicTacToe hypothesis and kept getting my hopes higher and higher.

Malice in Wonderland

Political discourse is often filled with exaggeration, both of the efficacy of one's own plans and of the deficiency of an opponent's plans.  Too often discourse also generates into calling into question the character and purpose of one's opponents.  And too often the rhetoric gets way off the mark.

I think Obama is engaging in wishful thinking that the various initiatives he proposes are going to create jobs.  Far too many of the jobs that match the "skill sets" of many workers are gone, not to return anywhere in the world.  Obama's opponents are engaging in wishful thinking that tax cuts will create jobs that match those "skill sets".  Sorry, folks, but investment is going to the jobs of tomorrow, not the jobs of yesterday.

Interestingly, both "sides" claim to be in support of the people.  But which people?  Republican John Weaver claims that victory will be given to the Republicans "by an out-of-touch, big spending president." ("GOP is riding a tiger in Tea Party movement", Star Tribune, 2010-09-16)  My question is which people is Obama out of touch with.  Certainly not all the people.  Almost all of the elections are a bit over one-third of the people voting for one party, a bit under one-third voting for the other party, and one-third of the people not voting at all.  Results may vary by electoral district.  Projecting results like these to a mandate of the "people" is living in Wonderland.

Bring freedom and democracy to who?

President George W. Bush has expressed many times his desire to bring "freedom and democracy" to other lands. His wishful thinking follows many other leaders before him.

Like other Great Powers before them, both Russia and the United States had to grapple with the hard fact that their "universalist" message would not be automatically accepted by other societies and cultures.

Paul Kennedy, The Rise and Fall of the Great Powers: Economic Change and Military Conflict from 1500 to 2000, pp. 394-395 in the chapter "Stability and Change in a Bipolar World, 1943-1980

For several other examples of expectations of "great" leaders not being met, see

When will they ever learn...