# Reviving the Ruins of Number Kingdom

The great wizard Factorial surveyed the empty buildings with an ever-increasing sense of dread. The staff of recursion he held was incredibly powerful and could do great damage in the wrong hands. In fact, it just had. One wave of the wand and you could use as many copies of a function as you like. Wave again, they all disappear. Wave it wrong, well…
A yell came from somewhere behind him. He, the standard operations, and the jesters had been searching the ruins for at least two hours. It was Division who had found the singular, shivering constant. All others had vanished.

Standard procedure in world-altering times is for the musketeers/operations to round up as many numbers as they can and protect them. Then if disaster strikes, the values can be used to construct all whole numbers from 1-100. Four numbers or more are usually enough to do the trick, and sometimes they get extra.

But one of the citizens of the number kingdom had gotten upset, and grabbed at Factorial's scepter. The only number spared was around ~2.6854, by sheer luck. Operations like Addition were rendered impossible without more than one value. However, the actions of Factorial the wizard, Sir Square Root, and the royal jesters Floor and Ceiling meant it was still possible to get an array of values.

#### *Can you get all values from 1-100, and possibly 101-120, using only factorials, square roots, and rounding, and a single instance of the luckless Khinchin's constant which lies between 2 and 3?*

Factorials can only be applied to integers. You will probably need factorials bigger than 170. This can be done using Desmos graphing calculator and a bit of exploration. Don't forget, if you get a number, you can use it as an intermediary for other numbers, e.g:
5040 = 7!
You don’t have to write out the expansion for 7 twice as long as you have it in your answer elsewhere. But you can't use 7+7 because that would use 2.6 twice. You could also write out:

$$\sqrt[2^{997}]{(166!)!}$$

to imply 997 square roots in a row. Even though you have neither a 2 nor a 997, you can still perform that operation because square roots use no numbers. However, you still need to achieve 166 somehow. You do not have to take either of these options; however, I highly recommend you take at least one, as the massive chains of square roots get unwieldy fast. I took the first option.

This puzzle came to be from this number formations puzzle. The inclusion of rounding functions inspired me to search for ways to get numbers using only the square-roots-of-factorials method. I already knew this method from Ian Stewart's Casebook of Mathematical Mysteries but the book only attempted so much because it could get more numbers by adding two of these numbers together. I had a lot of fun hunting down values, so instead of interrupting the current evolution of that problem, I created this question instead so others could try it out too.

• Hi, may I ask if you can use the double factorial such that $n!!=n(n-2)(n-4)...6\cdot4\cdot2$ if $n$ is even, and $n!!=n(n-2)(n-4)...5\cdot3\cdot1$ if $n$ is odd? Thanks! (By the way, I am on 51 out of 120 numbers right now, and I'll keep editing to add more.) Oct 21, 2022 at 10:25
• @CheeseCake I didn't need to :). On a different note, do you mind if I add some technical standard suggestions? Your answer is lagging me out Oct 21, 2022 at 14:47
• Thanks, I have fixed the stacks of square roots. Oct 22, 2022 at 4:36

I have solutions for all numbers from 1 to 1000. Each solution is represented as a list of operations to apply to the starting number in order, with s meaning square root, f floor, c ceiling, and ! factorial. So, for example, 4. c!!ssf represents the expression:

$$4=\left\lfloor\sqrt{\sqrt{\lceil K_0 \rceil!!}}\right\rfloor$$

Sequence of s longer than 2 are represented by a single s and a length (e.g. sssss becomes s5).

Here are the numbers from 1 to 120:

1. sf
2. f
3. c
4. c!!ssf!ssc
5. c!!ssf
6. c!
7. c!!ssf!sf!s3c
8. c!!ssf!sf!s3c!ssf
9. c!!ssf!sf!s3c!ssc
10. c!!ssf!sf
11. c!!ssf!sc
12. c!!ssf!ssc!!s4c!s5c
13. c!!ssf!ssc!!s4c!s5c!s3c
14. c!!ssf!sf!s3c!ssf!ssf
15. c!!ssf!sf!s3c!ssf!ssc
16. c!!ssf!ssc!!s4c!s5c!s3c!s3f
17. c!!ssf!ssc!!s4c!s5c!s3c!s3c
18. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c
19. c!!ssf!sf!s3c!sf!s6f!s5f
20. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4f
21. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c
22. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f
23. c!!ssf!sf!s3c!ssf!ssf!s3f
24. c!!ssf!ssc!
25. c!!ssf!sf!s3c!ssc!ssc
26. c!!sf
27. c!!sc
28. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f
29. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c
30. c!!ssf!ssc!!s4f
31. c!!ssf!ssc!!s4c
32. c!!ssf!sf!s3c!ssf!ssc!s3f
33. c!!ssf!sf!s3c!ssf!ssc!s3c
34. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f!s6f
35. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f!s6c
36. c!!ssf!sf!s3c!sf!s6f
37. c!!ssf!sf!s3c!ssc!ssc!s4f
38. c!!ssf!sf!s3c!ssc!ssc!s4c
39. c!!ssf!sf!s3c!sc!s6f
40. c!!ssf!sf!s3c!sc!s6c
41. c!!sf!s4c!s5c!s6f
42. c!!sf!s4c!s5c!s6c
43. c!!ssf!sf!ssf
44. c!!ssf!sf!ssc
45. c!!ssf!sf!ssf!s5c
46. c!!sf!s4f
47. c!!sf!s4c
48. c!!ssf!sf!s3c!sf!s6f!s4f!s9c
49. c!!ssf!sf!s3c!sf!s6f!s4c!s9c
50. c!!ssf!sf!ssc!s5f
51. c!!ssf!sf!ssc!s5c
52. c!!ssf!ssc!!s4c!s5c!s3c!ssf!s8c!s7c!s7f!s9f
53. c!!ssf!ssc!!s4c!s5c!s3c!ssf!s8c!s7c!s7f!s9c
54. c!!ssf!ssc!!s4c!s4f!s7f
55. c!!ssf!ssc!!s4c!s4f!s7c
56. c!!sc!s4f
57. c!!sc!s4c
58. c!!sc!s4c!s5c!s8c!s6f!s6f
59. c!!sc!s4c!s5c!s8c!s6f!s6c
60. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7f!s8c!s6c!s6f!s7f
61. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7f!s8c!s6c!s6f!s7c
62. c!!sc!s4c!s5f!s8f!s6f
63. c!!sf!s4f!s5f
64. c!!sf!s4f!s5c
65. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3f
66. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c
67. c!!ssf!sc!ssf!s6f
68. c!!ssf!sc!ssf!s6c
69. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f
70. c!!ssf!sf!s3c!sf
71. c!!ssf!sf!s3c!sc
72. c!!sf!s4c!s5c
73. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s3f!s8c!s7f!s10f!s7f!s9f!s8f!s10f
74. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s3f!s8c!s7f!s10f!s7f!s9f!s8f!s10c
75. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4f!s6c!s6f!s8f!s9f!s9c!s9c!s8f!s8f
76. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4f!s6c!s6f!s8f!s9f!s9c!s9c!s8f!s8c
77. c!!sc!s4c!s5c!s8c!s6f
78. c!!sc!s4c!s5f!s8f
79. c!!ssf!sc!ssf
80. c!!ssf!sc!ssc
81. c!!sc!s4c!s5c!s8c
82. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7f!s8f
83. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7f!s8c
84. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7c!s8c
85. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4f
86. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4c
87. c!!sf!s4f!s5f!s5c!s9c!s8f
88. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7f!s8c!s6f
89. c!!ssf!ssc!!s4c!s4f!s7f!s5c!s7f!s8c!s6c
90. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f!s6f!s4f!s8c
91. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f!s6f!s4c!s8f
92. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f!s6f!s4f!s8c!s6c!s7f
93. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6f!s4f!s6f!s4f!s8c!s6c!s7c
94. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c!s3f
95. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c!s3c
96. c!!ssf!ssc!!s4f!s4f!s6c!s9c
97. c!!sf!s4f!s5c!s5c!s9c!s9f!s10c
98. c!!sf!s4f!s5c!s5c!s9c!s9c!s10c
99. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c!s3f!s6c!s7c!s9f!s9f!s10c!s8c!s8f
100. c!!ssf!ssc!!s4c!s5c!ssf!s7f
101. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4f!s6f
102. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4f!s6c
103. c!!ssf!sf!ssc!s5f!s5f
104. c!!ssf!sf!ssc!s5f!s5c
105. c!!ssf!ssc!!s4c!s5c!ssc!s7c
106. c!!ssf!ssc!!s4f!s4f
107. c!!ssf!ssc!!s4f!s4c
108. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4c!s6f
109. c!!ssf!ssc!!s4c!s5c!s3c!s3c!s3c!s6c!s4c!s6c
110. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c!s3f!s6f!s7f!s9c!s9f
111. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c!s3f!s6f!s7f!s9c!s9c
112. c!!ssf!ssc!!s4c!s5c!s3c!ssf!s8c!s7c!s7f!s9c!s5c!s7f
113. c!!ssf!ssc!!s4c!s5c!s3c!ssf!s8c!s7c!s7f!s9c!s5c!s7c
114. c!!sc!s4c!s5f!s8f!s6f!s5c!s9f
115. c!!sc!s4c!s5f!s8f!s6f!s5c!s9c
116. c!!sf!s4f!s5f!s5c!s9c!s8f!s6f
117. c!!ssf!sf!ssc!s5c!s5f
118. c!!ssf!sf!ssc!s5c!s5c
119. c!!ssf!sf!s3c!ssc!ssc!s4f!s5f!s4c!s4c!s3f!s6f!s7c!s9c!s9c
120. c!!ssf!


The solutions up to 1000 can be found in this gist. Fun fact, 85,744! is the largest factorial that appears in those solutions.

So far I have revived $$89$$ out of the first $$120$$ numbers, with $$78$$ being in the first $$100$$.
I will keep editing. Please note that I have not used spoilers since this is too laggy - sorry for the inconvenience. Here is the partial answer:

$$\left\lfloor\sqrt{\sqrt{2.6854}}\right\rfloor=1$$
$$\left\lceil \sqrt{2.6854} \right\rceil=2$$
$$\left\lceil 2.6854 \right\rceil = 3$$
$$\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil= 4$$

$$\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor= 5$$
$$\left(\left\lceil 2.6854 \right\rceil\right)! = 6$$

$$\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil= 7$$

$$\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor= 8$$

$$\left \lceil \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rceil= 9$$

$$\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor= 10$$

$$\left\lceil\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rceil= 11$$

$$\left \lfloor\sqrt[2^5]{\left\{\left \lfloor \sqrt[2^3]{\left\{\left \lceil \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil\right\}!} \right \rfloor\right\}!}\right\rfloor= 12$$

$$\left \lfloor\sqrt[2^7]{\left\{\left\lfloor\sqrt[2^3]{\left\{\left \lceil\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rceil\right\}!}\right\rfloor\right\}!} \right\rfloor= 13$$

$$\left \lfloor \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rfloor= 14$$

$$\left \lceil \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil= 15$$

$$\left \lceil\sqrt[2^6]{\left\{\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil\right\} !} \right \rceil= 16$$

$$\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor= 17$$

$$\left \lceil\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rceil= 18$$

$$\left\lceil\sqrt[2^7]{\left\{\left \lfloor\sqrt[2^{13}]{\left\{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!\right\}!}\right \rfloor\right\}!}\right\rceil= 19$$

$$\left \lfloor\sqrt[2^4]{\left\{\left \lceil \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rceil\right\}!} \right\rfloor= 20$$

$$\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor= 21$$

$$\left \lceil \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rceil= 22$$

$$\left \lfloor\sqrt[2^3]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor=23$$

$$\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!= 24$$

$$\left \lceil\sqrt[2^2]{\left\{\left \lceil \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rceil\right\}!} \right\rceil= 25$$

$$\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor= 26$$

$$\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil= 27$$

$$\left\lfloor \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rfloor= 28$$

$$\left\lceil \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rceil= 29$$

$$\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor= 30$$

$$\left\lceil\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rceil= 31$$

$$\left \lfloor \sqrt[2^3]{\left\{\left \lceil \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil\right\}!} \right \rfloor= 32$$

$$\left \lceil \sqrt[2^3]{\left\{\left \lceil \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil\right\}!} \right \rceil= 33$$

$$\left \lceil\sqrt[2^7]{\left\{\left \lceil\sqrt[2^5]{\left \{ \left \lceil\sqrt[2^5]{\left \{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil \right\} !} \right\rceil \right \}!} \right \rceil\right\}!}\right \rceil= 34$$

$$\left \lfloor \sqrt[2^7]{\left\{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!\right\}!} \right\rfloor= 35$$

$$\left \lfloor \sqrt[2^6]{\left\{\left \lfloor\sqrt{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right\rfloor\right\}!}\right\rfloor= 36$$

$$\left \lfloor \sqrt[2^4]{\left \{\left \lceil\sqrt[2^2]{\left\{\left \lceil \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rceil\right\}!} \right\rceil \right \}!} \right \rfloor= 37$$

$$\left \lceil \sqrt[2^4]{\left \{\left \lceil\sqrt[2^2]{\left\{\left \lceil \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rceil\right\}!} \right\rceil \right \}!} \right \rceil= 38$$

$$\left \lfloor \sqrt[2^6]{\left \{ \left \lceil\sqrt{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right\rceil\right \}!} \right \rfloor= 39$$

$$\left \lceil \sqrt[2^6]{\left \{ \left \lceil\sqrt{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right\rceil\right \}!} \right \rceil= 40$$

$$\left \lfloor \sqrt[2^6]{\left \{\left\lceil\sqrt[2^5]{\left\{\left\lceil\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rceil\right\}!}\right\rceil \right\}!} \right \rfloor= 41$$

$$\left \lceil \sqrt[2^6]{\left \{\left\lceil\sqrt[2^5]{\left\{\left\lceil\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rceil\right\}!}\right\rceil \right\}!} \right \rceil= 42$$

$$\left \lfloor \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rfloor= 43$$

$$\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil= 44$$

$$\left \lceil \sqrt[2^5]{\left\{ \left \lfloor \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rfloor\right\}!} \right \rceil= 45$$

$$\left\lfloor\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rfloor= 46$$

$$\left\lceil\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rceil= 47$$

$$\small \left\lfloor\sqrt[2^{35}]{\left\{\left\{\left \lfloor\sqrt[2^7]{\left\{\left\lfloor\sqrt[2^3]{\left\{\left \lceil\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rceil\right\}!}\right\rfloor\right\}!} \right\rfloor\right\}!\right\}!} \right\rfloor= 49$$

$$\left \lfloor\sqrt[2^5]{\left \{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil \right\} !} \right\rfloor= 50$$

$$\left \lceil\sqrt[2^5]{\left \{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil \right\} !} \right\rceil= 51$$

$$\left\lfloor\sqrt[2^{43}]{\left\{\left\{\left \lceil \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil\right\}!\right\}!}\right\rfloor= 54$$

$$\left\lceil\sqrt[2^{43}]{\left\{\left\{\left \lceil \sqrt[2^2]{\left\{\left \lfloor \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil\right\}!\right\}!}\right\rceil= 55$$

$$\left \lfloor\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rfloor= 56$$

$$\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil= 57$$

$$\left\lfloor\sqrt[2^{11}]{\left\{\left\lfloor\sqrt[2^5]{\left\{\left \lfloor\sqrt{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right\rfloor\right\}!}\right\rfloor\right\}!} \right\rfloor= 58$$

$$\left \lfloor\sqrt[2^{20}]{\left\{\left\{\left \lceil \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rceil\right\}!\right\}!} \right \rfloor= 59$$

$$\left \lceil\sqrt[2^{20}]{\left\{\left\{\left \lceil \sqrt[2^2]{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right \rceil\right\}!\right\}!} \right \rceil= 60$$

$$\left\lceil \sqrt[2^6]{\left\{\left\lceil\sqrt[2^6]{\left\{\left\lceil\sqrt[2^8]{\left\{\left\lceil\sqrt[2^5]{\left\{\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil\right\}!}\right\rceil\right\}!}\right\rceil\right\}!}\right\rceil\right\}!}\right\rceil= 62$$

$$\left \lfloor\sqrt[2^5]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rfloor\right\}!} \right \rfloor= 63$$

$$\left \lceil\sqrt[2^5]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rfloor\right\}!} \right \rceil= 64$$

$$\left \lfloor\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rfloor= 65$$

$$\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil= 66$$

$$\left \lfloor\sqrt[2^6]{\left\{\left \lfloor\sqrt[2^2]{\left\{\left\lceil\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rfloor= 67$$

$$\left \lceil\sqrt[2^6]{\left\{\left \lfloor\sqrt[2^2]{\left\{\left\lceil\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rceil\right\}!} \right \rfloor\right\}!} \right \rceil= 68$$

$$\left\lfloor \sqrt[2^4]{\left\{\left\lfloor \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rfloor\right\}!} \right\rfloor= 69$$

$$\left \lfloor\sqrt{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right\rfloor= 70$$

$$\left \lceil\sqrt{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!} \right\rceil= 71$$

$$\left\lceil\sqrt[2^5]{\left\{\left\lceil\sqrt[2^4]{\left\{\left\lfloor \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rfloor\right\}!}\right\rceil\right\}!}\right\rceil= 72$$

$$\left\lceil\sqrt[2^6]{\left\{\left\lceil\sqrt[2^8]{\left\{\left\lceil\sqrt[2^5]{\left\{\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil\right\}!}\right\rceil\right\}!}\right\rceil\right\}!}\right\rceil= 78$$

$$\left \lfloor\sqrt[2^2]{\left\{\left\lceil\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rceil\right\}!} \right \rfloor= 79$$

$$\left \lceil\sqrt[2^2]{\left\{\left\lceil\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rceil\right\}!} \right \rceil= 80$$

$$\left\lceil\sqrt[2^8]{\left\{\left\lceil\sqrt[2^5]{\left\{\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil\right\}!}\right\rceil\right\}!}\right\rceil= 81$$

$$\left\lfloor\sqrt[2^{47}]{\left\{\left\{\left \lceil\sqrt[2^6]{\left\{\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil\right\} !} \right \rceil\right\}!\right\}!}\right\rfloor= 82$$

$$\left\lceil\sqrt[2^{47}]{\left\{\left\{\left \lceil\sqrt[2^6]{\left\{\left \lceil\sqrt[2^4]{\left\{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil\right\}!}\right \rceil\right\} !} \right \rceil\right\}!\right\}!}\right\rceil= 83$$

$$\left\lfloor\sqrt[2^4]{\left\{\left\lceil \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rceil\right\}!}\right\rfloor= 85$$

$$\left\lceil\sqrt[2^4]{\left\{\left\lceil \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rceil\right\}!}\right\rceil= 86$$

$$\left\lfloor\sqrt[2^3]{\left\{\left \lceil\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rceil\right\}!}\right\rfloor= 94$$

$$\left\lceil\sqrt[2^3]{\left\{\left \lceil\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rceil\right\}!}\right\rceil= 95$$

$$\small \left \lfloor \sqrt[2^6]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\lceil \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rceil\right\}!}\right\rfloor\right\}!} \right\rfloor= 101$$

$$\left \lfloor\sqrt[2^{13}]{\left\{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!\right\}!}\right \rfloor= 102$$

$$\left \lceil\sqrt[2^{13}]{\left\{\left\{\left\lceil\sqrt{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil}\right\rceil\right\}!\right\}!}\right \rceil= 103$$

$$\left\lceil\sqrt[2^5]{\left\{\left \lfloor\sqrt[2^5]{\left \{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil \right\} !} \right\rfloor\right\}!}\right\rceil= 104$$

$$\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor= 106$$

$$\left \lceil\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rceil= 107$$

$$\small\left\lfloor\sqrt[2^6]{\left\{\left\lceil\sqrt[2^4]{\left\{\left\lceil \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rceil\right\}!}\right\rceil\right\}!}\right\rfloor= 108$$

$$\small\left\lceil\sqrt[2^6]{\left\{\left\lceil\sqrt[2^4]{\left\{\left\lceil \sqrt[2^6]{\left\{\left \lceil\sqrt[2^3]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left \lfloor \sqrt[2^7]{\left\{\left \lfloor\sqrt[2^4]{\left\{\left\lfloor\sqrt[2^4]{\left\{\left\{\left\lceil\sqrt{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor}\right\rceil\right\}!\right\}!} \right\rfloor\right\}!} \right \rfloor\right\}!} \right \rfloor\right\}!} \right\rfloor\right\}!} \right\rceil\right\}!}\right\rceil\right\}!}\right\rceil\right\}!}\right\rceil= 109$$

$$\left \lfloor\sqrt[2^5]{\left \{ \left \lceil\sqrt[2^5]{\left \{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil \right\} !} \right\rceil \right \}!} \right \rfloor= 117$$

$$\left \lceil\sqrt[2^5]{\left \{ \left \lceil\sqrt[2^5]{\left \{\left \lceil \sqrt[2^2]{\left\{\left\lfloor\sqrt{\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!}\right\rfloor\right\}!} \right \rceil \right\} !} \right\rceil \right \}!} \right \rceil= 118$$

$$\left[\left\lfloor\sqrt{\left\lceil \sqrt{(\left(\left\lceil 2.6854 \right\rceil\right)!)!} \right\rceil}\right\rfloor\right]!=120$$

• You should hide your answer. See puzzling.stackexchange.com/editing-help#spoilers Oct 21, 2022 at 10:13
• @lukas.j Thanks, I already know, but it will take slightly too long. I can try and do it after I finished the answer. I'm on 51 numbers out of 120 now. Oct 21, 2022 at 10:16
• @CheeseCake I’m scared of all those square roots. You have enough floors to rebuild the ancient castles of Algebra, Geometry, and Calculus.
– user79541
Oct 21, 2022 at 11:23
• @someoneinexistence Hi, I have reduced the lagging by putting the symbol $\sqrt[2^n]{}$ instead of $\sqrt{}$ $n$ times. Oct 22, 2022 at 4:37
• This comment still poses problems when I attempt to view it on mobile - using my notation (with minor changes if you prefer to start directly from 2.6854) might help matters. Oct 24, 2022 at 16:31

Starting with $$3!!=720$$, the program below misses only (1-11 and) 14 other numbers below 1000 of which 530 is the lowest. (it only outputs how much factorials are used though, not the exact formula).

It takes the logarithm of factorial of numbers in the range $$(160-160^2)$$ and then the roots i.e divides by 2 until in that range again. (and then rounds)

Partial result.

No solution found yet for: 48, 49, 52, 53, 58, 59, 60, 61, 73 – 77, 82 – 84, 87 – 93, 96 – 99, 110 – 116, 119.

Only the numbers 2 and 3 are calculated from 2.6854. The others are calculated from a previously calculated number, so only showing the last step. For example for the number 5 it is:

 3: ⌈2.6854⌉
6: 3!         or: ⌈2.6854⌉!
26: ⌊√6!⌋      or: ⌊√⌈2.6854⌉!!⌋
5: ⌊√26⌋      or: ⌊√⌊√⌈2.6854⌉!!⌋⌋


Precedence based on faculty as a unary operator, so ! is applied before √.

Edit 2: 22 more found (only 13 to go)

58: ⌊√√√√√√√√√√√⌊√√√√√70!⌋!⌋
59: ⌈√√√√√√√√√√√⌈√√√√√70!⌉!⌉
82: ⌈√√√√√√√√√√√√⌈√√√√√75!⌉!⌉
112: ⌊√√√√√√√√√⌊√√√√√62!⌋!⌋
113: ⌈√√√√√√√√√⌊√√√√√62!⌋!⌉
114: ⌊√√√√√√√√√⌈√√√√√62!⌉!⌋
115: ⌈√√√√√√√√√⌈√√√√√62!⌉!⌉
119: ⌈√√√√√√√√√√√√√⌊√√√√√80!⌋!⌉
93: ⌊√√√√√√√√√√√√√√√√⌊√√√√√93!⌋!⌋
84: ⌈√√√√√√√√⌈√√√√√√√⌈√√√√√54!⌉!⌉!⌉
83: ⌈√√√√√√√√⌊√√√√√√√⌈√√√√√54!⌉!⌋!⌉
76: ⌊√√√√√√√√√√⌊√√√√⌊√√√√√42!⌋!⌋!⌋
77: ⌊√√√√√√√√√√⌈√√√√⌊√√√√√42!⌋!⌉!⌋
48: ⌈√√√√√√√√√⌊√√√√⌈√√√√√41!⌉!⌋!⌉
49: ⌈√√√√√√√√√⌈√√√√⌈√√√√√41!⌉!⌉!⌉
73: ⌊√√√√√√√√√√√⌊√√√⌈√√√√√38!⌉!⌋!⌋
74: ⌈√√√√√√√√√√√⌊√√√⌈√√√√√38!⌉!⌋!⌉
88: ⌊√√√√√√⌈√√√√√√⌈√√√√√48!⌉!⌉!⌋
89: ⌈√√√√√√⌈√√√√√√⌈√√√√√48!⌉!⌉!⌉
90: ⌈√√√√√√√√⌊√√√√⌊√√√√√√69!⌋!⌋!⌉
91: ⌊√√√√√√√√⌈√√√√⌊√√√√√√69!⌋!⌉!⌋
92: ⌈√√√√√√√√⌈√√√√⌊√√√√√√69!⌋!⌉!⌉

Edit 1: 8 more found

58: ⌊√√√√√√√√√√√⌊√√√√√70!⌋!⌋
59: ⌈√√√√√√√√√√√⌈√√√√√70!⌉!⌉
82: ⌈√√√√√√√√√√√√⌈√√√√√75!⌉!⌉
112: ⌊√√√√√√√√√⌊√√√√√62!⌋!⌋
113: ⌈√√√√√√√√√⌊√√√√√62!⌋!⌉
114: ⌊√√√√√√√√√⌈√√√√√62!⌉!⌋
115: ⌈√√√√√√√√√⌈√√√√√62!⌉!⌉
119: ⌈√√√√√√√√√√√√√⌊√√√√√80!⌋!⌉

Initial partial list:

1: ⌊√2⌋
2: ⌊2.6854⌋
3: ⌈2.6854⌉
4: ⌈√√5!⌉
5: ⌊√26⌋
6: 3!
7: ⌊√√√√√27!⌋
8: ⌈√√√√√27!⌉
9: ⌈√√7!⌉
10: ⌊√5!⌋
11: ⌈√5!⌉
12: ⌊√√√√√32!⌋
13: ⌈√√√√√32!⌉
14: ⌊√√8!⌋
15: ⌈√√8!⌉
16: ⌈√√√√√√57!⌉
17: ⌈√√√13!⌉
18: ⌊√√√√√√√102!⌋
19: ⌈√√√√√√√102!⌉
20: ⌈√√√√√36!⌉
21: ⌈√√√√22!⌉
22: ⌊√√√√√√108!⌋
23: ⌊√√√14!⌋
24: 4!
25: ⌈√√9!⌉
26: ⌊√6!⌋
27: ⌈√6!⌉
28: ⌊√√√√√√66!⌋
29: ⌈√√√√√√66!⌉
30: ⌊√√√√24!⌋
31: ⌈√√√√24!⌉
32: ⌊√√√15!⌋
33: ⌈√√√15!⌉
34: ⌊√√√√√√69!⌋
35: ⌊√√√√√√120!⌋
36: ⌈√√√√√√120!⌉
37: ⌊√√√√25!⌋
38: ⌈√√√√25!⌉
39: ⌊√√√√√√71!⌋
40: ⌈√√√√√√71!⌉
41: ⌊√√√√√√72!⌋
42: ⌈√√√√√√72!⌉
43: ⌊√√10!⌋
44: ⌈√√10!⌉
45: ⌈√√√√√43!⌉
46: ⌊√√√√26!⌋
47: ⌈√√√√26!⌉
50: ⌊√√√√√44!⌋
51: ⌈√√√√√44!⌉
54: ⌊√√√√√√√⌊√√√√31!⌋!⌋
55: ⌈√√√√√√√⌊√√√√31!⌋!⌉
56: ⌊√√√√27!⌋
57: ⌈√√√√27!⌉
62: ⌊√√√√√√78!⌋
63: ⌊√√√√√46!⌋
64: ⌈√√√√√46!⌉
65: ⌊√√√17!⌋
66: ⌈√√√17!⌉
67: ⌊√√√√√√79!⌋
68: ⌈√√√√√√79!⌉
69: ⌊√√√√28!⌋
70: ⌊√7!⌋
71: ⌈√7!⌉
72: ⌈√√√√√47!⌉
78: ⌈√√√√√√81!⌉
79: ⌊√√11!⌋
80: ⌈√√11!⌉
81: ⌈√√√√√√√√⌈√√√√√57!⌉!⌉
85: ⌊√√√√29!⌋
86: ⌈√√√√29!⌉
94: ⌊√√√18!⌋
95: ⌈√√√18!⌉
101: ⌊√√√√√√85!⌋
102: ⌈√√√√√√85!⌉
103: ⌊√√√√√50!⌋
104: ⌈√√√√√50!⌉
105: ⌈√⌈√⌈√⌈√⌈√⌈√⌈⌈(√⌈√12!⌉⌉!⌉⌉⌉⌉⌉⌉⌉
106: ⌊√√√√30!⌋
107: ⌈√√√√30!⌉
108: ⌊√√√√√√86!⌋
109: ⌈√√√√√√86!⌉
117: ⌊√√√√√51!⌋
118: ⌈√√√√√51!⌉
120: 5!

• I believe you are missing 3 square roots in 54 and 55 Oct 22, 2022 at 11:49
• Correct. Thank you very much for spotting that. Oct 22, 2022 at 11:52

Once we've found 1, 2, and 3 (which are trivial) there are essentially three kinds of moves we can make, which I denote as follows:

• !: a single factorial.
• Fn: a single factorial, followed by n square roots and a floor.
• Cn: a single factorial, followed by n square roots and a ceiling.

I wrote some code to, given an appropriate bound, do an exhaustive search within said bound using those steps and return a minimal path for each number. With intermediate values capped at 20000, my program finds all numbers between 3 and 999 inclusive, except 533, 640, 692, 701-2, 774, 788, 827, 855-7, 868, 874, 878, 898-899, 905, 908, 911, 929, 960, and 967-8. The results can be found in the accompanying text file because that's too much text to put here.
[EDIT: I have increased the cap to 33333, but I won't be able to push it higher without a significant upgrade. Now only 692, 772, 856-7, 868, 878, 908, and 960 are missing below 1000.]