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A proportional 36x54x54 barrel done with straight cuts. To prevent curved cuts many pieces were bandaged which transformed the puzzle into a 4x4x4.
This is a hexadecagonal prism with 9x36 sides.
In summer 2021 Ilya Toporgilka found one way to "play" a proportional 2x6x6 by allowing what he called "white olzing". He was glad that he achieved a proportional 2x6x6 as best he could.
After the 2x6x6 was "done", the builder tried to solve it (just once). Suddenly the smile from his face was gone. The puzzle was barely even movable. The inventor haven't known yet that a shapeshifting puzzle is not allowed to be made GIANT (like some of Tony Fisher's creations). If any puzzle of this size ends up being shapeshifted, it becomes impossibly hard to grip one of the layers to make a turn when there is such a mess of pieces sticking out. Ilya Toporgilka did not see any way of making a 2x6x6 smaller without too much pillowing. 11 mm cubies would have resulted in a size of a bacteria. Even a mini 2x2x2 has 12 mm cubies. Proportional double floppy cuboids (3x7x7 and so on) with "normal" sized cubies seemed explicitly impossible. No solution or even a clue was being found for a few months later.
The inventor was trying to connect double floppy cuboids with something. As he realised, it is not elliptic curves. He guessed it is similar to white nights in Saint Petersburg, but was he right?
Later Ilya Toporgilka tried to make a proportional 3x5x5 cuboid using the same technique that his proportional 4x6x6 used. His attempt resulted in a failure (a 5x5x5 was wasted). What could that be?
Serious thoughts followed then. Different combinations were being tested. Mostly Ilya Toporgilka was using techniques to build puzzles from the best architecturers from Germany, Netherlands, Great Britain and Spain. He received some help from Danish builder to build a 5x5x6 cuboid. For some reason Ilya Toporgilka have never thought about the USA and what architecture, mechanisms and techniques they have. To be continued.
It was regrettable for Ilya Toporgilka to discover that in the United States the food and mostly all packages come in gigaportions and on average bigger than in Russia and Europe. All food plates are much bigger than expected. It was a trifle for Ilya Toporgilka but it made him REALLY upset. He tasted a banana pie and it seemed enormous for him as the weight was 1150 grams (heavier than the mf8 petaminx) People are forced to eat such gigantic portions there. The food is mostly done "artificially". The organic products are a few times more expensive (in dollars/money), so on average they are harder to get. Technically healthy lifestyle is "less affordable" for "non-rich" people. It is impossible (or almost) to move to that country and not to become heavier in mass. Ilya Toporgilka estimates that even eating only american apples results in 5-7 kilograms gaining mass. Such facts Ilya Toporgilka just confirms. One can agree or disagree. But that is a fact. Like, for example, that more than 1 billion people live in China (much more than in most of the countries). Initially he had some of these thoughts in his head. Ilya Toporgilka did not claim it to be 100% truth but that was what started all the story!
After some thinking all the sudden the idea popped into the inventor's head. He then did some proofs and calculations. He was thinking hard what makes making smaller sizes so impossible or harder or "more expensive" sometimes. That lead to figuring out that there is a limit from the number theory! Ilya Toporgilka then made an equation and he found a few solutions in natural numbers. The biggest was a 36x54x54 made from a 4x4x4 by extending into a barrel and slitting every sticker into 9x9 squares. This puzzle is in fact a segment of a MUCH bigger theoretical puzzle(which is REALLY enormous in size).
The puzzle is based on number theory and Diophant equations.
Building a smaller portion of this puzzle cannot be done except a few of the examples. But they all turn out not too much cheaper(or even more expensive and giant).
Ilya Toporgilka says it took him 50 hours to build this. Almost. More than 40. At the end he stated: "It was worth it!"
Ilya Toporgilka supposes the puzzle can be built quicker if instead of extending to use the cubies from a few of the other 4x4x4s.
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Pentaemperor hexadecagonal prism
