Squeeze Me In Puzzles

Introduction

Squeeze-Me-In is a fun 2D puzzle where you’re trying to fit a bunch of pieces into a box! It begins with some pieces already inside the box and a smaller one outside. The goal is to move, flip, or rotate these pieces to fit everything inside without any overlaps. After squeezing in the smaller piece originally outside the box, the area of the assembly grows a little, but often subtly. There are lots of different versions of this puzzle, each with its own name, like Triangle+, Diamond+, and sometimes they even have irregular shapes. Think of them as a more challenging version of the classic Missing Square Puzzle, Square from Nowhere Paradox, and Infinite Chocolate Paradox.

Why designing new ones?

After learning about how this type of puzzle work (discussed later), I wanted to:

make easier version so beginners can also explore: many great one available commercially are fairly difficult and can frustrate beginners
make multiple challenges in one puzzle: like Squeeze-Them-In with different choices of the smaller pieces outside the box, so it is like multiple puzzles in one board.
share my designs in free and downloadable SVG files, so you can use them to 3D print or laser cut/engrave for yourself, family, and friends

What puzzles are you offering?

This is a list of growing calendar puzzles:
coming

How does it work?

Rectangular Grid + Stretched Polyomino (Polyrect)

There are different ways to make these puzzles. Let’s look at one using a rectangular grid. Imagine a 5x3 grid on the left, where each rectangle is taller than it is wide. This grid is nicely enclosed in a blue box. Now, let’s add another rectangle and rearrange them so their widths are bigger than their heights, creating a 4x4 grid, all surrounded by a red box, as you can see on the right.

Adding one unit into 15 unit (5x3) grid to form 16 unit (4x4) grid

The original 5x3 blue box on the left is also depicted as a blue dashed box within the 4x4 red box on the right. The red box is about 6.67% (1:15) bigger than the blue box due to the extra squeeze-me-in piece. The trick is that all the rectangular pieces have a 9:7 aspect ratio, which makes the blue and red boxes look similar in shape with small gap in between, as you can see in the figure above. It creates an illusion that adding the additional squeeze-me-in piece did not change the overall dimension.

Now, a puzzle with 16 identical rectangular pieces might not be very exciting. We need to make some interesting shapes. Here’s an example with 3 large pieces and the same small squeeze-me-in rectangular piece. When you’re making a physical puzzle, the playable rectangular opening on the board needs to be the same size as the red box so all 4 pieces can fit in. The original setup with just 3 pieces (without the squeeze-me-in piece) will fit into the playable box, but it will leave some small gaps. With rectangle base unit, these pieces are no longer polyominoes, but polyrects.

A simple squeeze me in puzzle at left (5x3 + 1) and its solution at right (4x4)

This idea works great for bigger grids and boards, like adding 1 to a 6x4 to get a 5x5, or adding 1 to a 10x9 to get a 13x7, as in this puzzle, as long as the aspect ratios are determined properly.

And guess what? If all the pieces in the puzzle have extra flip rules, we can even turn rectangular grid units and boards into parallelograms! Check out this puzzle as an example. This lets us transform the simple puzzle above into this one.

A parallelogram-shaped squeeze-me-in puzzle at left, and its solution at right

Rectangular Grid + Polyiamond and Polyabolo

Stay tuned

45° Rotated Board with √2-based Length

Here’s another way to establish the squeeze-me-in puzzle. Let’s take a look at this figure with a square grid and a grid size of 1, and we’ll use two boxes to show you the main idea.

Since 2√2 is almost 3, we can use above two boxes as the bases for our puzzle

The red box is 3 squares on each side, so it takes up 9 squares in total. The blue box is tilted at 45°, with each side measuring 2√2 (about 2.83). That makes its area 8 squares, which is just exactly 1 square smaller than the red box’s 9 squares. So, as shown below, it seems like the squeeze-me-in idea could work as shown below!

squeezing 1 square to a 2√2x2√2 board (8 squares) to form a 3x3 box (9 squares)

Next, we need to find pieces that can make this happen. However, the edge of the 45° tilted blue box crosses the square units diagonally, which means we can't use polyomino pieces to establish the puzzle. Instead, we use polyabolo (also called polytan) pieces. As an example, here is a set of five polyabolo pieces that fit into the blue box as shown below.

A Simple Squeeze-Them-In Puzzle

It’s fascinating that we have two options for the squeeze-me-in piece: a square with an area of 1, or a parallelogram with the same area. Each can be individually squeezed into the large red box along with original five polyabolo pieces, making it a two squeeze-me-in puzzles - we can just call it a squeeze-them-in puzzle !

With smaller boxes and boards, this example might not seem too intriguing. However, as we increase the size of the board, like squeezing 1 square to a 7x7 board (49 squares) to create a 5√2x5√2 box (50 squares), or squeezing 2 squares to a 7√2x7√2 board (98 squares) to form a 10x10 box (100 squares), the design becomes much more interesting and challenging to solve.

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