Latin Squares, Squared Squares, and Legoed Squares

I introduced my kids to Latin Squares the other day. If you know Sudoku then you have seen examples of Latin squares. The idea is to fill in a grid of squares with colors or numbers or some other symbols, such that each symbol appears exactly once in each row and each column. (see also the Wikipedia Latin square entry). I handed them graph paper and let them loose. This is what my eight-year-old produced:
Note that she produced some Latin squares, but also explored other ideas.

I wasn’t sure initially how long my kids would be content to explore Latin squares, so I also planned to tell them about squared squares. A squared square is just a square made up of smaller squares (this is easy to accomplish). A perfect squared square is much more difficult, consisting of squares each of different sizes. For a long time it was thought that perfect squared squares didn’t exist, but they do! Here is the smallest (in some sense) one:
We got excited about it and decided to make one for our wall:

Next, the ten-year-old wanted to produce some squared squares of his own, but found the perfect ones difficult (they were thought to be completely non-existent for a long time after all). He settled for producing imperfect ones. After making an imperfect squared square on graph paper, he decided to reproduce it with Legos. Here is the result: This led us to considered perfect Legoed squares. That is to say, squares made up of some number of Lego pieces, each piece having a unique size. Here is one of our first examples: It’s a 3×3 made up of a 1×1, a 1×2, and a 2×3. Here’s another, somewhat larger: We found a bunch more (maybe ten or so total). I’m not sure that we got all of them. How many can you find?

Look for up-coming posts on: 1) further progress in building our marble computer, and 2) teaching my five-year-old to count (which required some deep thought about just what counting is).

Rep-tiles

It’s been a while since I’ve done an evening of math with my two older kids (6 years and 8 years), so this week I decided I would teach them about rep-tiles. (If you don’t know what rep-tiles are, don’t worry.  I’ll be explaining that shortly, or check out this excellent article on rep-tiles)

First we pulled out our copy of the board game Blokus (a great game involving placing polyominoes).  I laid out the four monominoes on the board and asked them if they could make a square of the four.  “Of course!”

I had told them previously that we would be working with rep-tiles, not the animals, but something to do with mathematics.  They had been dying to know what a mathematical rep-tile was.  I asked them if now they understood.  “It’s a shape that you can make a bigger one out of a bunch of them.”  I was surprised that they caught on so quick, expecting something more along the lines of “a shape that you can make a square from several copies of it.”

I had told them previously that Utah was a rep-tile (The pentomino that most people call “P”, I call “Utah”, me being from Utah originally).  So we did that one next.  It took them several minutes to piece four copies of Utah together into a bigger one and they passed the pieces back and forth several times.  Here is what they came up with:

I had them select several other polyominoes and decide whether they were rep-tiles.  The domino, for instance, was easy.  The tetromino Zstumped them.  With the aid of paper and a pen I gave them an argument that Z is not a rep-tile.  They were convinced and later gave me a similar argument that the pentomino Xis not a rep-tile (how could you fill in a corner with a smaller copy?).

I told them that of course not only polyominoes can be rep-tiles.  For instance what about circles? “No,” they said.  Packing circles together always leaves gaps, after all.  I had previously cut out some polyiamonds (also see: another polyiamond article) printed on card stock.  We proceeded to determine which of these were rep-tiles.  We spent quite some time on this, but I’ll skip the details and give you the end result. Namely, we pasted our rep-tiles to construction paper.  The kids and I were pretty pleased with the results.