A room measuring 5 by 5 is tiled with a checkered design.
If you started in the upper left corner and walked along the perimeter in a
clockwise manner, you would step on exactly 8 gray tiles. Suppose you walked
around the outside edge of a larger square room, with the same checkered floor
design, and you stepped on exactly 2006 gray tiles.
What would the dimensions of the room be?
Your answer must be supported by a sufficient amount of work.
Solution 1 by Andrew Jeanguenat, De La Salle Collegiate High School
I solved this problem by solving the formula first for the total number of tiles
that touch the perimeter of a square checkerboard. Let S = the length of one
side. I used the side of a tile on a checkerboard for 1 unit. I figured out
the formula for the number of tiles on the perimeter by finding the area of
the checkerboard and then subtracted the area of the smaller checkerboard inside
the original one. Therefore, the number of tiles on the perimeter 
Exactly half of the tiles on the perimeter are gray. So, there must be 
gray tiles. If there are 2006 gray tiles on the perimeter, then 
;
so 
. The dimensions
of the checkerboard are 1004 by 1004.
Solution 2 by Lots of Students
Many students considered a variety of examples and observed that the number
of gray tiles t and the side length s had the following relationship,
. Solving the equation,
, they discovered
that 
. Therefore,
the dimensions of the room would be 1004 by 1004.
Also solved by: Ben Thomas (Adrian H.S.); Tim Wu (Ann Arbor Huron H.S.); Robert Harrison, Ian McDowell (Lansing Catholic Central H.S.); Ahmedov Ikbof (Benzie Central H.S.); Brent Nixon, Kim Penzien, Craig Rogers, Kyle Sinko (St. Louis H.S.); Jake Maloney (Detroit Catholic Central H.S.);Jackson Smith (Caro H.S.); Elisabeth Bramble, Mary Hogan, Martina Kuehn (Marshall H.S.); Pranav Moudgil (Detroit Country Day H.S.); Shawna Wei, Jay Yang (Forest Hills Central H.S.); Derek Banagis, Allison Bryan, Ted Burgoun, Joe Davila, Max Dixon, Joe Mester, Caroline Sierra, Joe Sugivama, Dani Tyner, Marc Walker (Forest Hills Eastern H.S.); Paige Bennett, Brian Siebers (Forest Hills Northern H.S.); Haley Jackson, Allison Krueger, Michelle Marigha, Katie Nelson, Kayla Niezgoda, Sarah Stompin (Ladywood H.S.); Kim Lebioda, Dan Nadratowski, Walter Wang (Plymouth H.S.); Matthew Brownell, Armila Francis, Cameron Groenewoud, Patrick Lutz, Katie Wendt (Plymouth Christian Academy); and Matt Baumgarther, Swapna Gudipati, Jacqueline Gutierrez, Nathan Helmkamp, Melanie Miller, Kiran Sebastian (Saginaw Heritage H.S.).