Impossible rectangle?

Cut out the two identical, imperfect rectangles shown here—each missing two corners. Follow the lines to divide them into four geometric pieces… Then try to reassemble them into one perfect rectangle.

Sounds simple? Think again! Solve it? Tell us what made it such a brain-bender!

I’ve always had a passion for puzzles made of simple geometric pieces—especially those that seem almost impossible to solve despite the deceptively simple shapes and limited number of elements. As an Op Art artist, I find these visual enigmas a delight not only for the eye but also for the mind. For someone drawn to minimalism like me, beauty lies not just in pure form, rules, or apparent simplicity, but in the very intention of the game: to create something concrete and well-defined out of very little. And yet, at first glance, the pieces rarely seem to match the information at hand—as if something’s always missing, or as if the pieces resist aligning with your will.

Back in the ’80s, I created numerous puzzles with these paradoxical traits—some even became worldwide hits. When people would say, “Ah, so you’re the creator of that devilish puzzle?” I would always reply, “No, not a puzzle, but a piece of optical art.” Or: “No, not a puzzle, but a visual paradox.” Or sometimes: “No, not a puzzle, but a moment of zen-like reflection.”

No, I’ve never created puzzles—but rather works that turn geometry into visual meditation.

⇨ More visual enigmas to create.

The Trapezoid Trap

Here’s a rather tricky puzzle—perfect for the classroom or a fun activity with your kids (and possibly an excuse to sharpen your own spatial skills).

Print and cut out the five puzzle pieces (see Fig. A), then try to fit them all into the larger shape (Fig. B) without overlapping. Yes, it’s possible… As you may have noticed, all the pieces—including the final shape—are similar right trapezoids. They do, however, vary in scale, just to keep things interesting.

Cut out the 5 puzzle pieces (right trapezoids; fig. A) in order to fit them all into the larger shape (fig. B) without overlapping.

You can download the full template in PDF format here.

The first person to post a correct solution will receive a set of our original postcard designs.

And if you find yourself strangely fascinated by these slanted quadrilaterals, you’re not alone. Even ancient Greek mathematicians toyed with shapes like these to explore proportions and symmetry. Curious about trapezoids (or wondering if they’re secretly out to get you)? Here’s a helpful read: https://en.wikipedia.org/wiki/Trapezoid

Happy puzzling—and remember, if it feels impossible, you’re probably getting close.

(Hint: Some pieces may need to be flipped over, as if seen through a mirror.)

World Map on a Dodecahedron

With the holiday season approaching, here’s a fun and educational activity to enjoy with your kids. Assemble a three-dimensional world map by cutting and folding a single-piece dodecahedron template featuring a gnomonic projection by Carlos A. Furuti.
Download the PDF template here.
A simple and creative way to explore geography while spending quality time together.

Intriguing Geometric Dissections

Any two polygons with equal area can be dissected into a finite number of pieces to form each other. One of the most interesting dissection or geometric equidecomposition puzzles is that discovered by Harry Lindgren in 1951 (see Fig. 2 further below). As you know, in this kind of puzzles, the geometric invariant is the area, since when a polygon is cut and its pieces are distributed differently, the overall area doesn’t change.

Lindgren was the first to discover how to cut a dodecagon into a minimal number of pieces that could pave a square, when rearranged differently. His solution is very elegant, he first built a regular Euclidean pavement by cutting a dodecagon as shown in fig. 1.a, then arranging the four pieces symmetrically on the plane (fig 1.b). The tessellation achieved with these pieces corresponds, by superposition, to a regular paving of squares. The example in fig. 2 shows how the puzzle appears once finished.

dodecagon fig. 1

Continue reading “Intriguing Geometric Dissections”

Rowboat Dilemma

A rowboat is floating in a harbor, and a stubborn donkey pulls by mean of a long rope through a pulley the boat toward the shore. When the donkey has moved 1 meter, how far has the boat moved:
a) exactly 1 meter,
b) more than 1 meter,
c) less than 1 meter?

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show solution