Registone Mechanism

The Registone Mechanism is an ancient difference engine created by the Regipastries in the early 1880s. It sits in the sacred Shrine of the Waffle, which was later used to store an ingredient of the Golden Waffle, in the Dorkugese Jungle. It is made of nothing but rocks throughout, the only non-mineral item being its output device/printer. Even the ink itself is made from rocks, that is, charcoal.

Description
The Regipastries were an ancient sect of nerds obsessed with machinery. They were at least sixty years ahead of their time in their vision that machinery would one day be able to do math and help people. Yet, they didn't fully understand what a computer could do.

The Regipastries devised the Registone Mechanism to act as a calculator. It could take any polynomial function (i.e. those weird-looking $$f(x) = 2x^2 + 3x - 7$$ things) and calculate any of its output values (called the range), using simple addition and subtraction. All it needed to start was a piece of paper giving it initial information and some patient Regipastry to turn the crank. The mechanism could predict their oversimplified economy and changes in the miserably small stock market, for starts. In fact, it could predict the outcome of about any process that involved a mathematical pattern, albeit VERY slowly.

They used what they had. The console is nothing but rocks, in a cave of rocks. Rock gears, rock cogs, a rock crank, carved rocks for numbers, rock spools to hold the paper (not rock), rock workings, ink from charcoal (also a rock), and it was set on a rectangular slab of rock as well. Since rocks erode over time, the Registock Mechanism epically failed when it came to reliability. One hundred Regipastries carved deeper into the cave to permanently bunk with their precious machine, to maintain and operate the machine. Over the decades, they began to die off, and, in 2010, there were only seven left at the Shrine.

Meanwhile, other members of the tribe built two other difference engines, the Regifrozitron and the Regiron Console, and the tribe continued to print numbers and eat waffles undisturbed, for many years after.

After the advent of the telegraph, Regipastry priests could venture out of their villages and make contact in populated areas to other members of the tribe, using a special phone line dedicated to them. There are seven of these hotlines in existence. Their existence is classified from the Governance and is wiretapped and maintained by a secret group of EPF elites. In the past forty years, the hotline has been used eleven times. Regipastry members use more modern technology; the hotline is exclusively for utmost business.

Gigaregibot
In 1971, the now-dying tribe of Regipastries, having long succeeded in their past three endeavors, decided to pool their power one last time and build the ultimate security console, in conjunction with the kings and queens of Olde Antarctica, in case they or the government ever needed an above-maximum security vault of the highest priority.

Like the other two consoles, the Gigaregibot requires a printed sheet from each engine to compute. If the sheets are correct, the Gigaregibot will activate the vault mechanism and open. From the Registone, one must compute the first thirteen range values for the function $$f(x) = x^2$$ -- that is, square 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13.

Golden Waffle
In exchange for advice on how to carve rocks more efficiently and allow for less wear and tear, Leonardo da Waffli was permitted to store an ingredient to the Golden Waffle in the shrine, inside the Registone Mechanism. The ingredient may be accessed by telling the Registone Mechanism to compute a certain, secretive function that will cause it to activate a vault mechanism, inside which the ingredient is stored.

Trivia

 * This is a parody of the Pokemon, Regirock, a creature made of nothing but rock (hence then name) and one of three protectors of Regigigas.
 * The console was never seen by non-Regipastry eyes until IBM, Explorer, and Fred discovered it.