MATH JOKES YAY

Educating through the Game Fifteen

Nine cards are marked with digits 1 through 9 and placed face-up on a table between two players. The players take turn picking one card at a time. The winner is the first player to get the sum of exactly 15 from among any three of his or her cards.

  

  This game was proposed by our professor Marteen during our first meeting, as an early morning mind-exercise.  It was fun and quite a work out for brain, but he made it harder for us by demanding what he called as “the winning strategy.” Basically, he asked us whether or not there is a strategy that can ensure us to always win this game, and if such strategy exists, to find it. We worked in group of 5 people, and none of us could come up with that kind of strategy. I did note the following cases though:

1.    We made a list of all possible combinations of numbers that sum up as 15, in which 5 has the most frequent appearance. So we used that as strategy and always started with 5, but it did not ensure us to always win the game.

2.   There was this check-mate like situation in which whatever cards your opponent pick, you will always win. For example, I start first and I pick 8. You pick 4. Then I choose 6. You definitely have to pick 1 to prevent me from winning with 8+6+1=15. Then I choose 2. Now I will always win since I have 2 opportunities: 8+2+5=15 and 6+2+7=15, while you only have 4 and 1. The problem is to find out how to recreate such situation. 

3.    When our opponent picked a certain card, it was almost automatic for us to do quick calculation and pick a card that would prevent them for getting the sum of 15, therefore blocking them. It was only when Marteen brought it to our attention that we realized that it was not us being smart, but our opponent controlling us. 

It turned out that there is a winning strategy, only slightly went out of what it is defined to be.

MAGIC SQUARES AND TIC TAC TOE

First thing though, I am not digressing. This actually has something to do with what I was talking about. 

Magic square is an arrangement of numbers (usually integers) in a square grid, where the numbers in each row and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number.  The size of a magic square is denoted by its number of columns and rows; magic square with 3 rows and 3 columns is said to be of size 3x3. There are 9 grids and the numbers in each row and in each column as well as the numbers in the diagonals, all add up to 15.

It is easy to see that playing this game here is like playing tic-tac-toe on a magic square.  Now we just have to ensure that we can always win in a tic-tac-toe.

DA WINNING STRATEGY: DOES IT EXIST?

Winning strategy in general means just like what Marteen said. But in tic-tac-toe, there are no sure-win strategies; a winning strategy is simply one that will maximize the chance of a player winning (Yeo, 2012). So you will either end up a winner or in a draw. Some winning strategies for tic-tac-toe follow. 

Imagine a tic-tac-toe game where player 1 uses X and player 2 uses O.

Case 1: player 1 starts in the center. The game will always end in a draw, assuming each player makes optimal moves.

Case 2: player 1 put the first x in a corner. If player 2 does not put the first O in the center, they will lose. For example, if player 2 put the first O in the corner opposite of the first X, player 1 can place second X in any of the remaining corners. There are two choices right? Player 2 will be forced to put an O in between two Xs to prevent player 1 from winning. Player 1 can win by placing an X in the last corner.

It is easy to figure out how to win if player 2 put the first O in any for the edge squares.

Case 3: player 1 put the first x in a corner. If player 2 put the first O in the center, there is still a way to win, but only if player 1 puts the second X at the corner opposite the first X and player 2 puts the second O in any of the remaining corners.  Then player 1 can win by placing the third X in the last corner.

Case 4: player 1 put the first X in an edge-square. If this is the case, then the game is wide open. Either player can win, although the game will most often end in a draw. Below are example of this case.

Therefore, the winning strategy for player 1 is to place the first X at the corner. For player 2, pray that player 1 will not start at the corner and if they do, put the first O in the center. 

Back to game we were talking about before, it is clear that any player can win the game by following this tic-tac-toe strategy on a magic square. That is why we never won when we started with 5, because 5 is located at the center. Although it will take extra effort since the players must visualize the magic square; otherwise, opponents will be aware of the strategy.

WHAT DOES IT HAVE TO DO WITH MATH EDUCATION?

Yeo (2012) introduced this game as Fifteen and stated that it can help students learn mathematics and develop heuristic skills such as examining all possible scenarios and systematic listing, spatial visualization, and thinking skills, including predicting, conjecturing, generalizing and checking. The question is, to children of what age should we introduce this game? 

This game is exciting because it requires the players to work his best and develop a strategy to win the game. But not every student will employ that particular mental action against the game. How children respond to games is apparently linked to their developmental levels. Kamii and Nagahiro (2008) summarize these developmental levels as follows. 

- Level 0: the child does not try to win. They are focused on doing what they are supposed to do, here by getting a sum of 15.
- Level 1: the child tries to win. They know they are supposed to get a sum of 15 and do it before the other player does.
- Level 2:  the child tries to block the opponent.
- Level 3: the child is at the most advanced level; they compare the consequences of winning the game now with blocking the opponent or foresee that the opponent has two possible ways of winning.

Most 4-year-old are at level 0, while 6 and 7-year-olds are generally at the highest level. Therefore, it will best to introduce this game to students in 6^th grade or higher. 

Aside from all those technical matters, we teachers know that most students love to play games and from the beginning I already though that this game will be a fun addition in the classroom. Fifteen is an interesting and thought-provoking game that helps students learn mathematics at the same time.

Sources:
Yeo, J.B.W. 2012. "Fifteen: Combining Magic Squares and Tic Tac Toe". Mathematics Teacher, vol 106, no. 1. USA: NCTM.

Kamii, et. al. 2008. "The Educational Value of Tic-Tac-Toe for Four to Six-Year-Olds". Teaching Children Mathematics. USA: NCTM.

Review: the Act of Killing (2012), From an Indonesian's Perspective

Director : Joshua Oppenheimer
Genre    : Documentary
Rating   : 95% (Rotten Tomatoes)

I have heard about this movie and the hype surrounding it months ago, but did not have the will to download and watch it, thanks to the endearing internet speed of Surabaya. But then Jack Harries recommended it and I knew I just have to watch it.

ermahgerd look at all the awards

 The Act of Killing is a documentary movie about the chaotic genocidal period following the coup d’etat against Indonesia’s first president and founding father, Soekarno, during the 60's. It is directed by Joshua Oppenheimer and nominated for last year Academic Award for Best Documentary.  Yep, the dark history of my motherland is now bared all for the whole world to see – well, the whole world who bother to see this kind of stuff. It follows Anwar Congo, one of the main executioners of PKI, as he describes and reenacts the memories of his past life as a ‘jagal’ (butcher) together with his old friends and affiliates. For your information, PKI is how people used to call the communist party back then, as well as marginalized people and Chinese ethnics. 

That’s it? Well, basically yes. Oppenheimer himself said that this a 'documentary of imagination' - we are going to see nothing that directly depicts all that murders and massacres, aside from Congo’s recalling, retelling and reenactment. 

The first thing that crossed my mind is now I know what German people feel. I know about G-30S/PKI (that’s how we call it – since it is started at September 30, 1965), we did learn about it in grade school, but like Pak Adi (one of the character in the movie) said, the history is told and written from the winner’s point of view. Back then, I only had the history book our teacher gave us. But then I grew up and started to have interest in history. I read a lot about World War 2, African Civil War, and other stuff, as well as movies, but never once it crossed my mind to open the history book of my own country. 

G-30S/PKI, there is no other way to say it, is a violation of human rights. It is cruel, sadistic, illogical, and unfair. It is basically mirror image of what has been happening throughout the world since the dawn of the humankind; one group of people think they are right, better and above the rest, therefore they have the right to judge, punish, kill and well, basically act as a god. I am talking about the people doing the killing here; in the grand scheme of things, greater reason may exists. But these people, regardless of what these massacres actually evolved from, definitely do not think that what they did is wrong. 

Now, years later, after the media are no longer doing their role to bias our opinion, we can look back with clean logic and know that it was wrong. But they can’t. They are standing in the winner box of the history and clearly do not feel like getting out anytime soon. Every sick thing they do is not only right; it is respectable, heroic and even honorable. It is just so disturbing the way they boast about what they did without even a slightest hint of regret.

Also, they are so stupid. Non-Indonesian might not know this since it is not recognizable with English subtitle, but when you are Indonesian and you have to listen to the way they talk and see the way they carry themselves … god. 

Especially the way they treat the Chinese ethnics. For all of you who don’t know, Chinese people have been the black sheep of our society since God knows when. Most Chinese people in our country totally fulfill their stereotype as hardworking, good with counting, and stingy people. They, of course, are rich as hell. Indonesians, of course, envy them. What did we do when we envy people for their money? Definitely not working as hard as or even more than them. We hate them, that’s what we do. And from the moment we hate people, all reason to  segregate, mistreat and banish them from the universe suddenly seems fair. 

Yay. Isn’t it awesome being a part of people like this.

You know what, I am Indonesian and I am fully aware of how outsiders perceive us as a nation. I used to be surprised and a little bit pissed off because well, I did not feel that way. Of course, we are not Netherlands-perfect, but we are doing well; I’m not hungry or being forced to quit school and marry someone 40 years older. But watching this movie, it hits me so hard that maybe my country is actually just as much of a third-world country as everyone think; I just do not see it because I have been living in a frickin buble.  

I thought my homeland, the place where I was born, is better than this piece of shit. 

But you know what, maybe I am wrong. Maybe my country IS better than this. Who knows right? I am feeling this way because I am watching this movie, and this movie is created by a director; a human. He took control on what he put in the movie and who knows if he told 100% truth. We all know the power of media. All that matter is everything comes back to the single individuals. You can’t just bunch people into one bundle of trash and throw it into the garbage. Nice people exist and even though they might never be able to fix a country, at the very least they won’t turn into one of those mindless criminals. 

The movie end with Congo being eaten by the nightmares of his past life and having a mental breakdown so severe he retches on the floor where he used to kill his victims. One thing I like about this movie is that the director is kind enough to put a human face on top of this unspeakable, shameful crime. Congo’s regret definitely healed me a bit from all those traumas.

Trivia: Joshua Oppenheimer is a foreigner but can speak Indonesian. He acts as interviewer throughout the movie. One of the co-director is credited to an anonymous Indonesian. In fact, most of the crew is credited as anonymous.The film is not granted release in Indonesia, for fear that it might provoke something in society. None of the people I know knows about the existence of this movie.

Spooky isn't it.

Chocolate Oatmeal Mug-Cake

When I first moved to Utrecht, I was pretty excited about our kitchen. It’s a standard middle-class European kitchen, but for me it’s awesome. It has microwave, oven, fridge, stove – I usually had to make do with just a stove. 

To be noted though, cooking is not really my hobby. I just like doing it. I also don’t like cooking food that takes a long time to make, that’s why I tend to pursue recipes that are fast and easy.

Now, meet my Chocolate Oatmeal Mug Cake.

It doesn’t look very appealing, I know, but it taste pretty good. This is especially for breakfast, because it is fulfilling and easy to make. Basically, it’s a regular oatmeal-and-milk version 2.0. The recipe is very flexible, you can make substitute and use everything according to your liking. Basically, try with equal parts of wet and dry stuffs, and then adjust accordingly. In this recipe I used half a mug of milk and half a mug of oatmeal. 

1.    Wet stuff: Get a nice big mug. Crack in an egg and half a mug of your preferred milk. Mix well with fork.

2.   Dry stuff: Get another cup. Combine half a mug of oat, a quarter teaspoon of baking powder, a pinch of salt, a teaspoon of cocoapowder and sugar, if you like your breakfast sweet. You can adjust the amount of sugar according to your taste, but use more than what you use when you make oatmeal, because the egg fade out the sweetness a lot.  

3.    Mix the stuff: Add half the dry mix to the wet mix, stir well, then add the rest and stir again. Think cake batter, not cookie mix – if it looks too thick, add another splash of milk. 

4.    Cook the stuff: Put it in the microwave. I put mine in my lunch box, and it takes 3 minutes to go from a cup of goopy ingredients, to a Cup-of-Awesome, slightly chewy on top, soft and cake-y through the middle, and soft and fudge-y on the bottom. You can also put some Nutella in the mix, so when its cooked you’ll get melted chocolate in the middle of your cake.