Loose-Packing Puzzle

The solution of a loose-packing puzzle of filling 4 Z pentominoes into a red square frame

Introduction

A loose-packing puzzle is a tiling/packing puzzle where the solution of packed pieces doesn’t fill the entire frame, leaving some gaps. These gaps are often between the pieces and the frame border, or there can be internal gaps (holes). As you can see in the solution of a loose-packing puzzle in the image above, the 4 Z pentomioes form a compact region, but they don’t fill the entire red square, leaving some gaps with the red frame.

In the puzzles we discussed before (such as Calendar, Replication), the “grid” or orientation of the pieces is easy to identify from, for example, the frame border. For a loose-packing puzzle, it might not be straightforward to see it. Again, using the 4 Z pentominoes puzzle as an example, before putting the 4 pieces into the square frame, how do you orient the Z pieces when placing them in? Often, our intuition would align the Z pieces with the square frame and try to put them at the corners or lie flush against the sides, like below:

An attempt to solve the puzzle by aligning the orientations of the pieces and frame

But this does not solve the puzzle. The solution, as you can see in the first picture of this post, requires a 14° turn to the packed region. Now you can see why those loose-packing puzzles can be tricky—they need the perfect mix of both the right packing and the right way to orient everything.

Martin's Menace Puzzle

Martin's Menace (originally known as Four Fit) is a fantastic example of 2D loose-packing puzzles. The challenge is to fit four pentominoes into a rectangular frame. But don’t let its simple appearance fool you! The right combination of packing and orientation is a real puzzle to solve. If you’re feeling tired, you might find a solution in a YouTube video

What puzzles are you offering?

• Three Fit, Two Times - an easier version of Four Fit, using only three pentomino pieces, but with two frames to solve
• Pentomino Heart - pack six different pentomino pieces into a heart-shaped frame
• Hex P & Heart - pack eight tetrahex P pieces into a heart-shaped frame
• Eight T Challenge - pack four or eight identical T pentomino into two squares

Creating this type of puzzles with just one solution can be tricky because we need to consider every possible way the grid can be oriented and moved. The worst part is when pieces jump from one grid to another, which is hard to capture. To illustrate this, here’s an example

The image on the left illustrates how three pentominoes (F, W, and Z) can be arranged within a rectangular frame. Notice that all three pieces are positioned on a rotated polyomino square grid, tilted about 27° from the frame. Computer simulations indicate that there’s no other configuration that fits these three pieces into the frame using a grid with different rotation angles and starting points. This suggests it could be a promising loose-packing puzzle with a single solution. However, as you can see in the image on the right, there’s actually another solution where F and W pentominoes are on a different grid than the Z pentomino. Comprehensive computer simulations to prevent designs with this type of solution is not available yet, and thus manual inspection is still required.