We always try to push the limits, to get the minimum or maximum arrangements that a puzzle set can yield. One question we were always curious to find: What is the maximum number of border spaces that can be filled in the Poly-5 tray with just the pentominoes? We knew it could not be 100%; there just aren't enough pieces to stretch the whole length. We'd gotten close with just two non-pentominoes helping to fill out the border. Then Sara Hopper, on September 15, 2011, in our pavilion at the Maryland Renaissance Festival, came up with this record-breaking beautyjust one outside piece. Hurray, Sara!
In October 2012, Doug Caine sent us this slightly modified version that even has all colors grouped:
If we count the one non-pentomino as occupying 5 unit edges of the perimeter, is there a way to get it down to 4? Can you do better or prove the solutions shown above are, indeed, the best result possible? A prize for such an answer! Email your solutions to Poly-5 Perimeter.
Another rare solution is to have the smallest pieces form a square in the center. Here is one example, though one orange corner is touching it.
Here's the square off-center, floating in a sea of the pentominoes, and the colors symmetrically grouped, solved by Eric Bare:
The further question is, can the small square be centered and surrounded entirely by only a single color? Yes, and George Sicherman found it and won a prize for this beautiful solution. Again, the colors are grouped.
Two other tricky challenges were the tetromino center line (left, in blue) solved by Meshele Merchant, and the smaller pieces separated around the perimeter (right), by Eric Bare. It's interesting that these solutions were found for the first time after more than 25 years of our making this puzzle. There is no end...
Nathan Gilbert set a new record with this solution that has only 4 unit edges that are not pentominoes. For good measure he also grouped the interior pieces. Wow, Nathan!