Chessboard Tetris Mathematics

It’s time for the Power ARML, which is one of my favorite math contests. The atmosphere within the math room is rife with collaboration and ideas. Today’s power round topic is Tetris. Most of the team is excited because many of them are familiar with the game dynamics and techniques.

We delegate the problems between members, and the scramble begins. After 30 minutes out of the allotted 40, we are done with almost all of the power round problems.

The team is waiting on me for a few crucial calculations that will be important for another problem. However, time is running out and I need to find a more practical way of solving my problem. Brute force approaches will only get you so far.

The question was:

3b. If the puzzle is ten spaces wide, there are 10C2 = 45 different patterns where two squares remain on the bottom line and all other squares have disappeared. Some of these patterns can be achieved with just one row disappearing while others require at least three rows to disappear. How many of each kind are there?

I listed out all possible combinations and found the answer. But to my horror, the next question proved to be much tougher:

If you try to repeat part (b) of the previous problem but without using the “T”-shaped block, you will discover that some patterns still require only one line to disappear, others require three lines to disappear, but now some are impossible altogether.

4a. Determine how many of the 45 patterns still require only one line, how many require three, and how many are now impossible.
4b. Give a simple explanation why some patterns are impossible without the “T”-block.

Source: Spring 2014 ARML Power Contest

The seven different tetrominos used in Tetris. The “T”-block refers to the purple tetromino.

I tried thinking about what distinguished the T-block from the other tetrominos. Suddenly, thoughts of chess and different pieces wandered into my mind. I had to stop getting distracted – there wasn’t a whole lot of time left. But wait a minute, chessboard… black and white – parity! It all came together, it was parity! If you put all the tetrominos on a chessboard, a T-block covers three squares of one color and one of the other, while the other tetrominos have an equal split of color.

I quickly wrote down my solution and shared it with the other team members. We extended my parity argument and casework and floated through the other remaining problems with little time to spare.

My chessboard epiphany made me realize that solutions could come from anywhere. I just have to look in the right places for innovative ideas to approach different problems.

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