Quadratum Cubicum

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Lately, I have been fortunate enough to have tried some of the most amazing puzzles, since I started collecting them, back in 2008. The latest addition to my collection is the Quadratum Cubicum, which was entered at last year's 31st IPP Design Competition.

The Quadratum Cubicum is a group of nine square trisection puzzles, ranging from six to nine pieces, that essentially, turn a bigger square into three identical smaller ones. Christian Blanvillain was the one responsible to group together the eight best solutions from past mathematicians and turn them into a unique puzzle.

As you might have have noticed, I mentioned that there were nine solutions in the Quadratum Cubicum, but only eight were chosen. That's because the remaining one and most recent solution, was discovered by Christian himself in 2010. His solution, however has a particularity, as it's the first one proposed, that has all its six pieces having the exact same area. One of my favorites, because of its elegant symmetry.

(Click to Enlarge) - Christian Blanvillain's Solution - 2010

In history, there have been many examples of how to cut a square into three identical squares. Abu'l Wafa' was the first one, in the tenth century, to have come up with a solution to this problem, with nine pieces. Also, one of my favorites. To know more about the history of the square trissection, visit this page.

(Click to Enlarge) - Abu'l Wafa' - 10th Century

Others have followed since Abu'l's solution. Nobuyuki Yoshigahara, the famous Japanese puzzle designer, known for his many Hanayama Cast design contributions, have also discovered one solution in 2003, with nine pieces and actually, the only one with the same piece arrangement for the same three squares. Curiously enough, it was the hardest one for me to solve. Another notable mathematician, the French Édouard Lucas, famous for inventing the classic Tower of Hanoi puzzle, also proposed his solution in 1883, a century before I was born.

(Click to Enlarge) - Left: Édouard Lucas - 1883; Right: Nobuyuki Yoshigahara - 2003

Now, to the Quadratum Cubicum puzzle itself...

The puzzle is presented in a beautifully made wooden box with pin-lock, measuring 18.5cm x 18.5cm x 3.5cm (7.28" x 7.28" x 1.38"). Inside, comes with the nine solutions, with its 68 pieces, all built in plexiglass. The material chose for the pieces looks very appealing, although after some use, you might see them covered in fingerprints. A good wipe with a microfiber cloth will make them as new, though.

One thing that I appreciate the most in puzzles, and very few can provide it, is the variety of challenges that you can try and solve with them. The Quadratum Cubicum is fortunately, one of such examples and throws at you a few of them, with different levels of difficulty.

(Click to Enlarge) - Left: Enry Perigal - 1891; Right: Greg N. Frederickson - 2002

With the 68 pieces of the nine solutions combined, there are four sizes of squares that you can build: 1 large square with 42cm (16.53") in length; 3 medium large squares with 24.2cm (9.53") in length; 9 medium small squares with 14cm (5.51") in length and finally, 27 small squares with 8.1cm (3.19") in length. By the way, the logo on the box, cleverly shows this notable relation between the square sizes and numbers.

The puzzle already comes with its nine solutions, solved into their medium small squares, although by no means, it will make things easier for you. The first challenge that you can try to solve, in order to become more familiarized with all the solutions and try for the bigger square, is to solve each of the nine solutions, into its smaller three identical squares. When you turn all nine solutions into smaller squares, you'll be left with 27 squares. By stacking them on top of each other, you can build a cube with exactly the same height as the length of a small square, since each piece has the thickness of 3mm (0.12"). This cube is simply called the Cubicum.

(Click to Enlarge) - Cubicum w/ 27 Squares

After you've solved the solutions into the 27 squares, you can try next to solve them into their original squares, one at a time. Some of them will be easy, but you'll find others much more tough. The ones I found the hardest, were the Nobuyuki Yoshigahara's and the Edouard Lucas'.

Building the next larger three squares, even though its counter-intuitive, is actually much easier, because you just have to place nine smaller squares from three solutions, side-by-side in a 3x3 grid.

(Click to Enlarge) - Medium Large square

On the other hand, building the larger 42cm square, is the hardest challenge. In principle it's easy to build it, because like the above example, it uses all nine 14cm squares in a bigger 3x3 grid. The hard part, however, is to solve it. By mixing all 68 pieces, it becomes extremely challenging to recognize which piece belongs to what  solution. That's why, in order to successfully solve it in a good time frame, you have to know most of the nine solutions by heart. It took me almost a whole afternoon to finally have the biggest square built.

The first three or four solutions were relatively easy to solve, because their pieces were easily recognizable, but the rest were quite challenging and given the fact that some had very similar pieces, complicated things much more. Slowly, with time, I got to finish another two or three solutions and, as the number of pieces got progressively smaller, so it became less complicated, until with much relief of mine, there was only one solution left. I had finally conquered the ultimate Quadratum Cubicum challenge.

(Click to Enlarge) - All 9 Solutions and 68 pieces, solved into the largest square

When you're done with the Quadratum Cubicum challenges, there are a few more that you can try to solve. Building a rectangle, using the highest number of pieces that you can and without using any of the nine solutions, is probably the most interesting one. But there are more... Visit this page to know more.

Picpulp, the company that builds the Quadratum Cubicum, has a few other designs presentations for this puzzle, that can fit anyone's budget. Other than the Collector's Box you see in this review, there's another one, also with the nine solutions, but with the box format of the Cubicum, the cube with the 27 squares. You can also get a box with just three solutions of your choice, or if you prefer to just try a single solution, there are more affordable options. Finally, if you have one of the three boxes with three or more solutions and want to solve one solution at a time, you can get a separate tray, just for this task.

(Click to Enlarge) - Left: Abu Bakr Al-Khalil - 14th Century; Middle: Colonel De Coatpont - 1877; Right: Same as left one

Closing Comments:

From the history behind the solutions, to the actual presentation and build quality of the puzzle and the variety of challenges, Christian Blanvilain's Quadratum Cubicum is a very special addition to any puzzle collection, and the different levels of difficulty, as well as the varied purchasing options, sure makes it accessible to anyone, puzzler or not.

I haven't been so immersed in a puzzle for quite some time, for the Quadratum Cubicum is the dream puzzle of the packing and dissection fans and most importantly, it's like owning a mathematical treasure.

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