The nice wizard Factorial surveyed the empty buildings with an
ever-increasing sense of dread. The workers of recursion he held was
extremely highly effective and will do nice injury within the mistaken palms. In
reality, it simply had. One wave of the wand and you would use as many
copies of a operate as you want. Wave once more, all of them disappear. Wave it mistaken, properly…
A yell got here from someplace behind him. He, the usual
operations, and the jesters had been looking out
the ruins for no less than two hours. It was
Division who had discovered the singular, shivering fixed. All others had
vanished.

Standard process in world-altering instances is for the musketeers/operations to spherical up as many numbers as they will and shield them. Then if catastrophe strikes, the values can be utilized to assemble all entire numbers from 1-100. Four numbers or extra are often sufficient to do the trick, and generally they get further.

But one of many residents of the quantity kingdom had gotten upset, and grabbed at Factorial’s scepter. The solely quantity spared was round ~2.6854, by sheer luck. Operations like Addition have been rendered not possible with out multiple worth. However, the actions of Factorial the wizard, Sir Square Root, and the royal jesters Floor and Ceiling meant it was nonetheless potential to get an array of values.

Can you get all values from 1-100, and probably 101-120, utilizing solely factorials, sq. roots, and rounding, and a single occasion of the luckless Khinchin’s fixed which lies between 2 and three?

Factorials can solely be utilized to integers. Don’t neglect, if you happen to get a quantity, you should utilize it as an middleman for different numbers. You will in all probability want factorials larger than 170. This may be performed utilizing Desmos graphing calculator and a little bit of exploration.

^{This puzzle got here to be from this quantity formations puzzle. The inclusion of rounding features impressed me to seek for methods to get numbers utilizing solely the square-roots-of-factorials technique. I already knew this technique from Ian Stewart’s Casebook of Mathematical Mysteries however the ebook solely tried a lot as a result of it might get extra numbers by including two of those numbers collectively. I had plenty of enjoyable searching down values, so as an alternative of interrupting the present evolution of that drawback, I created this query as an alternative so others might attempt it out too.}