OMSC 18

 It’s the 18th Annual Ottawa Mathematics and Statistics Conference! 

For the last few editions of the OMSC, we thought it would be fun to design a themed logo around the mathematical properties of that number edition. For the 16th edition, we had a colorful grid of squares – four by four, of course. The next year, we went for a slightly more obscure fact. Did you know that 17 is the only prime number that is the sum of four consecutive primes? This gave us the idea to make the logo a stylised Young diagram for this cool partition. Now, we are thrilled to be hosting the 18th edition of the OMSC! The trouble is, what makes the number 18 special? Unfortunately, it’s not some amazing prime. Its partitions are (in our opinions) mediocre. We had to go even more obscure than last year… 

Did you know there are 18 unique pentominoes? These are polygons made of 5 equal-sized squares connected edge to edge (think Tetris, except Tetris uses tetrominoes).

But how to use these for our logo? We had to find some exciting tiling. Conveniently, there is an entire area of combinatorics dedicated to polyomino tiling! There are lots and lots of pentomino tilings and packings on a variety of surfaces, but we can up the ante by requiring additional constraints around symmetry and colouring. In researching this topic, we found a great example of exactly this type of puzzle. 

Can we tile a rectangle using all 18 one-sided pentominoes so that they have a proper 3-coloring satisfying the following properties: 

1. “The six pentominoes that are symmetric over reflection are the same colour.” 

2. “The other twelve pentominoes are coloured with the remaining two colours so that the pentominoes that are reflections of each other are different colours?” 

*Spoiler Alert*: This is impossible. (Feel free to give it a shot or find a proof that it cannot be done. If you do find a proof, we would love to see it!) 

It turns out that we can have such a colouring on a tiling of a different shape that is almost a rectangle. Check out our logo above! How many different shapes can you find that admit a tiling with these constraints?