[Ghoser777]

A little background - I'm a math teacher, so I'm use to studying things that seem to have no purpose, so bare with me here

One of the other teachers in my department invented (or stole eloquently) this game called Math Tic-Tac-Toe and has been having his students play it on crazy days (10 minute classes? Yeah, lots can get done). Here's how the game works:

1. You have a standard Tic-Tac-Toe board in a 3x3 grid.
2. A card will have numbers in each of the boxes. "Smart" players will use only numbers in the range of 2 to 12 (although some never figure it out).
3. The teacher rolls two dice.
4. If the sum of the two values of the dice match a number on your card, you can put an "X" through it (only one number per turn!)
5. The first player with a winning Tic-Tac-Toe board (3 in a row, 3 in a column, 3 in a diagonal) wins (ties aren't broken).

So the questions we've been throwing back and forth is what the best Math Tic-Tac-Toe board is. I'm curious what you think. For those who don't want to see my guess first, I'll put it in white below:


6 3 9
4 7 10
5 11 8

[ManOfSteal]

2 4 6
3 5 7
8 10 12

[Ghoser777]

Originally Posted by ManOfSteal

2 4 6
3 5 7
8 10 12

Hmmm, mine seems to win 85% of the time, yours 25% of the time (sometimes we tie). I like mine Maybe I'll throw an applet together so people can test it out on their own (my numbers come from generating 1000 different games and seeing who wins first with random rolls of the die).

Question - why use 12 when you cane use 11? Why use 2 when you can use 9?

[ManOfSteal]

Originally Posted by Ghoser777

Question - why use 12 when you cane use 11? Why use 2 when you can use 9?

Not sure that I can pass along a statistical answer to you...what are my winning odds if I include the 11 and 9 to my original grid?

9 4 6
3 5 7
8 10 11

[Ghoser777]

Significantly better! I'm winning only 68% of the time, while you're winning about 53% of the time.

[ManOfSteal]

gorickey's board:

7 6 4
3 10 9
5 8 11

[Ghoser777]

Interesting statistics - I'm winning 73% of the time, while you're winning about 54% of the time. I think that means you're winning more often, but more of those wins are ties. Interesting...

[beadman]

Applying a bit of logic:
The center square should have the 7 in it, since 7 is the number with the highest probability (6/36), and the center square is part of 4 out of 8 possible wins. The next highest numbers of possible wins are the four corner boxes - each is part of three possible wins. The least number are the center top and bottom, and the center left and right - each are only part of two possible wins out of 8.

Therefore, the 6 and 8 should be in corner boxes (each has 5/36 odds), the 5 and 9 should be in the other corner boxes (each has 4/36 odds). The remaining boxes should have the 4 and 10 (3/36 odds) and the 3 and 11 (2/36 odds).

This leaves the question: should the 6 and 8 be diagonally opposite, or should each have a 5 or 9 diagonally opposite like this:

6 10 5
3 7 11 or
9 4 8

6 10 8
3 7 11
5 4 9

If the second, then should the middle top be the 10, which has a probability of 3/36, or the 3, which has a probability of 2/36? In other words, should the worst probability numbers "partner" with the highest, or not?

beadman

[Ghoser777]

I find it interesting that no one has suggested duplicates yet...

[Ghoser777]

beadman:

Your first card seems to be pretty much equivalent to mine - yours is a reflection and a rotation of mine, plus the 4 and the 10 are switched. What's interesting about that is because 4 and 10 have equal chance of showing up, switching their location shouldn't change the true probability. This pans out, as our winning percentages are both roughly 86%.

Your second card is slightly inferior to both your first and my card - I win about 69.8% of the time while yours wins 69.5% of the time. The difference is so small that it probably could be discarded. One thing of note - your second card doesn't have a 6-7-8 combo, which is the highest probable winning combo you can have without duplicates!

[beadman]

I agree the second card is slightly inferior - and it also does not have the 6-7-8 combo. I was curious if it would make much difference. It's also interesting that you mention the duplicate idea. You didn't specify that no duplicates were permitted, so... Hmmmm, how about a card all 7's in it?

beadman

[Xeo]

So basically you're playing bingo on a 3x3 instead of 5x5 and picking the numbers a different way.

I don't have a guess for your question, but considering some numbers are more probable than others, seems that bingo is a more fair game since each number has virtually equal probability. Then again, maybe I misunderstand and the kids pick their own numbers. If that's the case, then yeah, that is an interesting game to put into a math class, although probably more stats than high school kids can handle. Unless you're at a university?

Anyway, I'm just blabbering.

[Ghoser777]

Ah, that's what everyone in my department thought first! (the all 7's). Duplicates are allowed, yes. The problem is, if you have all 7's, think of all the redundant winning combinations you have! You only need one row, column, or diagonal of 7's to capture all the times the all 7's board would win, plus you get all the other ways you could win with non-7's. In fact, I've never found a good board with any duplicates, and that's one thing I'm trying to explore - is there ever a time where duplicate numbers helps your probability instead of hurts it?

[DeathMan]

just make your simulator run every combo a statistically relevant number of times, and see which work the best.

[beadman]

Originally Posted by Ghoser777

... is there ever a time where duplicate numbers helps your probability instead of hurts it?

So, you're saying that has a lower "pay off" than your original setup?

How about something like

7 6 7
8 7 6
7 8 7

BTW, I was just playing around with your idea and tried using three dice - makes a nice 4 by 4 layout with the numbers 3-18. Since there's no single center box, and no single number with highest probabilty (10 and 11 are each 27/216), it makes some interesting looking tic-tac-toe cards. So far, the best I've come up with, using no duplicates, is having the 10 and 11 in diagonally opposite outer corners and the 9 and 12 on the same diagonal on the inside - 10, 9,12,11 from top left to bottom right, in other words...

beadman