four fours

A semi-popular mathematical puzzle/truth which states that all the integers between zero and 100 can be described using four fours and assorted mathematical symbols. For instance:

0 = (4+4)-(4+4)
1 = (44)/(44)
2 = (4*4)/(4+4)
3 = (4+4+4)/4
4 = sqrt(4*4)*(4/4)
5 = sqrt(4*4)+(4/4)
6 = (4!/4)+4-4
7 = (44/4)-4
8 = 4*4-4-4
9 = 4+4+(4/4)
10 = (4/.4)-4+4

One's effectiveness at solving these puzzles as the integers get larger depends in large part on one's creativity and mathematical vocabulary. Or else their ability to program a computer to do the dirty work for them.

In eighth grade, Mr. Benzing gave our honors pre-algebra class this assignment: we were to come up with one equation equalling (or rounding down to) every number from one to 100, using any mathematical operators/punctuation we knew of and exactly four 4s. We had a week or so.

I was working with my good friend Alexis. We only really worked on this project during class, when we were supposed to be paying attention to whatever lesson Mr. Benzing was trying to drill into our adolescent heads.

After a couple days, Lex and I got pretty frustrated with this little exercise. We found solutions for the majority of numbers, but there were a select few that we couldn't get. Anyway, it sure felt like a cop-out on Mr. Benzing's part — stick the class with this damn-near-impossible assignment, and kick back with a beer every night for a week instead of grading dull homework. Now, I realize that the assignment encouraged creativity and problem-solving skills, but for every brain cell of mine it stimulated, it infuriated two more.

We didn't get an equation for every number (but, as I recall, neither did any other group in the class).

This assignment ended up frustrating Lex and I so much that we started looking upon the number 4 as a manifestation of pure evil, devised solely to aggravate and haunt us both. We denounced the assignment, and massaged our hurt egos by impugning the work of everyone who got further than we did.

---

A couple years later, I find myself sitting in an honors geometry class headed by the unbelievably dull Mrs. Krill. Geometry had always come easy to me (it's even more obvious than algebra), as well as to Jim, who sat next to me. Jim had also been in one of Mr. Benzing's honors pre-algebra classes, and remembered the Four Fours assignment as vividly (and with as much disgust) as I had.

Of course, being mischevious young sophomores without any intentions of paying attention to the class material at hand, we dug through our old notebooks until we discovered the list of equations we had turned in. We compared our solutions, discovered we were stuck on the same handful of equations, and started trying to come up with the missing links.

Before long, our search had led to results equalling a good portion of the numbers from 101 to 200. Having no limit on free classtime, we decided to one-up Mr. Benzing and try to discover equations for the numbers up to 200.

Here is what we came up with.

1. 4 − √4 − ⁴⁄₄
2. 4 + 4 − 4 − √4
3. 4 − √4 + ⁴⁄₄
4. 4 + 4 − √4 − √4
5. 4 + √4 − ⁴⁄₄
6. 4 + 4 − 4 + √4
7. 4 + 4 − ⁴⁄₄
8. 4 + 4 + 4 − 4
9. 4 + 4 + ⁴⁄₄
10. 4 + 4 + 4 − √4
11. ⁴⁄_.4 + ⁴⁄₄
12. 4 + 4 + √4 + √4
13. 4 + √4 + ⁴⁄₄
14. 4 + 4 + 4 + √4
15. (4 × 4) − ⁴⁄₄
16. (4 × 4) + 4 − 4
17. (4 × 4) + ⁴⁄₄
18. (4 × 4) + 4 − √4
19. 4! − 4 − ⁴⁄₄
20. 4! + 4 − 4 − 4
21. 4! − 4 − ⁴⁄₄
22. (4 × 4) + 4 + √4
23. 4! − √4 + ⁴⁄₄
24. (4 × 4) + 4 + 4
25. 4! + √4 − ⁴⁄₄
26. 4! + √4 + √4 − √4
27. 4! + 4 − ⁴⁄₄
28. 4! + 4 + 4 − 4
29. 4! + 4 + ⁴⁄₄
30. 4! + √4 + √4 + √4
31. 4! + √4 + ^√4⁄_.4
32. 4! + ⁴⁄_.4 − √4
33. 4! + 4 + ^√4⁄_.4
34. 44 − ⁴⁄_.4
35. (^{(4! + 4)}⁄.4)⁄_√4
36. 4√4 + 4! − 4
37. 4! + ^{(4! + √4)}⁄_√4
38. 4! + ⁴⁄_.4 + 4
39. 44 − ^√4⁄_.4
40. 44 − √4 − √4
41. ^{(4! + √4)}⁄_.4 − 4!
42. 44 − 4 + √4
43. 44 − ⁴⁄₄
44. 44 − 4 + 4
45. 44 + ⁴⁄₄
46. 44 + 4 − √4
47. 4! + 4! − ⁴⁄₄
48. 44 + √4 + √4
49. 44 + ^√4⁄_.4
50. 44 + 4 + √4
51. ^(4!–√4)⁄_.4 − 4
52. 44 + 4 + 4
53. 4! + 4! + ^√4⁄_.4
54. 44 + ⁴⁄_.4
55. ⁴⁴⁄.4⁄_√4
56. 4! × √4 + 4 + 4
57. ^{(4! − √4)}⁄_.4 + √4
58. 4! + 4! + ⁴⁄_.4
59. ^4!⁄_.4 − ⁴⁄₄
60. 44 + 4 × 4
61. ^4!⁄_.4 + ⁴⁄₄
62. ^4!⁄_.4 + 4 − √4
63. ^(4!–√4)⁄_.4 − √4
64. ^{(4√4 × √4)}⁄₄
65. ^4!⁄_.4 + ^√4⁄_.4
66. 4 × 4 × 4 + √4
67. ^{(4! + √4)}⁄_.4 + √4
68. 4! + 4! + 4! − 4
69. ^{(4! + √4)}⁄_.4 + 4
70. 4! + 4! + 4! + √4
71. ^{(4.4 + 4!)}⁄_.4
72. ^(4!√4)⁄_{(4 + 4)}
74. 4! + 4! + 4! + √4
75. ^{(4! + 4 + √4)}⁄_.4
76. (⁴⁄_.4)√4 − 4!
80. 4 × 4! − 4√4
81. (^√4⁄_.4 + 4)√4
82. ^4!⁄_.4 + 4! − √4
84. 44 × √4 − 4
86. ⁴⁴⁄_.4 − 4!
87. 44√4 − .4
88. 44 + 44
89. 4! + ^{(4! + √4)}⁄_.4
90. 4 × 4! − 4 − √4
91. 4 × 4! − ^√4⁄_.4
92. 4 × 4! − √4 − √4
93. (4! × 4 − √4 − .4)
94. 4! + ^{(4! + 4)}⁄_.4
95. 4 × 4! − ⁴⁄₄
96. (⁴⁄_.4)√4 − 4
97. 4 × 4! + ⁴⁄₄
98. (⁴⁄_.4)√4 − √4
99. (4! × 4 + 4 − 4)
100. ⁴⁴⁄_.44
101. 4 × 4! + ^√4⁄_.4
102. 4 × (4! + √4) − √4
104. (4! × √4 + 4)√4
105. ^{(44 − √4)}⁄_.4
106. ⁴⁴⁄_.4 − 4
108. ⁴⁴⁄_.4 − √4
109. ⁴⁴⁄_.4 − .4
110. ^{(4! + 4! − 4)}⁄_.4
111. ⁴⁴⁴⁄₄
112. ⁴⁴⁄_.4 + √4
114. ⁴⁴⁄_.4 + 4
115. ^{(44 + √4)}⁄_.4
116. ^{(4! + 4!)}⁄_.4 − 4
118. ^{(4! + 4!)}⁄_.4 − √4
119. ^{(4! + 4!)}⁄_.4 − √.4
120. ^{(44 + 4)}⁄_.4
121. (⁴⁴⁄₄)√4
122. ^{(4! + 4!)}⁄_.4 + √4
124. ^{(4! + 4!)}⁄_.4 + 4
125. ^{(4! + 4! + √4)}⁄_.4
126. ^4!⁄.4⁄_.4 − 4!
128. (^4!⁄_.4 + 4)√4
130. ^{(4! + 4! + 4)}⁄_.4
134. ⁴⁴⁄_.4 + 4!
135. 4!√44 − 4!
140. ^(4!√4)⁄₄ − 4
142. ^(4!√4)⁄₄ − √4
144. ^{(4! + 4!)}⁄_.4 + 4!
146. ^4!⁄.4⁄_.4 − 4
148. ^4!⁄.4⁄_.4 − √4
149. ^4!⁄.4⁄_.4 − .4
150. ^4!⁄.4⁄_.4 + .4
152. ^4!⁄.4⁄_.4 + √4
153. (4! × 4 × 4 × .4)
154. ^4!⁄.4⁄_.4 + 4
155. (4!√44 − 4)
156. √((4 + 4)!) − 44
157. 4!√44 − √4
158. 4!√44 − .4
159. 4!√44 + .4
160. ⁴⁴⁄_.4 − 4
161. 4!√44 + √4
163. 4!√44 + 4
164. 4!√44 + √(4!)
168. (^4!⁄_.4 + 4!)√4
170. ^{(44 + 4!)}⁄_.4
172. 44 × 4 − 4
174. 44 × 4 − √4
175. ^{(4! + 4)}⁄.4⁄_.4
176. 44√4 × √4
178. 44 × 4 − √4
180. 44 × 4 + 4
183. 4!√44 + 4!
184. (4! × 4 − 4)√4
188. 4!(4√4) − 4
190. 4!(4√4) − √4
191. 4!(4√4) − .4
192. 4!(4√4) + .4
194. 4!(4√4) + √4
196. 4!(4√4) + √4
200. (4! × 4 + 4)√4

Credit where credit is due: All the above formulas were happened upon by either Alexis Sachin or myself, with these notable exceptions:
Benjamin Korin: 119, 135, 153, 155, 156, 157, 158, 159, 161, 163, 164, 172, 174, 178, 180, 183
James Perkins: 176, 188, 190, 194, 196
Travis Schoen: 37
Kyle Parker: 56

I think these are all correct, but if you find one that's wrong please let me know.

I've made a point not to go searching the Internet for the solutions I'm missing — it must be a pride thing. The assignment was given before most people had email addresses, nevermind fast Internet access, so since I couldn't look up the answers then, I'm not going to now.

However, if you've discovered a solution for one of the numbers I'm missing (through your own invention, please, and not through a Google search!), please /msg me and help my soul rest easy.