Questions on the Grid
AIM Math Teachers' Circle
23 July 2012

These are some questions made up by the participants of the 2012 AIM Math Immersion Workhop after looking at a blank piece of grid paper for a few minutes.

Can you create a game using dots, combining Conway's game of Life, and
Blockus, to make it a multi-player game?

How many rectangles are there?

Folding on the lines, what is the largest volume open box you can create?

How many ways can you put rectangles of a particular size onto this
grid, such as 1x1 or 1x2 rectangles?  (Using some of each,
potentially?)

How many ways can we divide it in half?  Symmetrically or not?  By
drawing a continuous boundary along the grid lines.

How many pentominoes are there, and will any of them exactly fill the
grid paper?

Beginning in the top left corner square, moving down or right, how
many paths are there to the lower right square?

With the paper in landscape, if you drop a Plinko chip from the top,
moving one square left or right each time it moves down, what are the
possible squares in the bottom where it could land and what is the
probability of each?

How many perfect squares are on the page?

If you colored in each square using 9 different colors, without any
same-color squares touching, how many squares of each color would
there be?