Construct the Integers 1-100 using Six 5s Solution

Objective:	Produce the integers 1-100 using only combinations of 6 fives and the mathematical operations: +, –, *, / and !.
	You may adjoin 5s together, so the number 55 is a possible combination of 2 5s.
	You may also use the operation !!, a double factorial. (Note: 5!! = 5*3*1 = 15.)
Note:	I am still looking for a way to express the numbers 64 and 88 using only the operations +, –, *, / and !.
	If you find a way, please send me an e-mail at wgarner [at] ucsd [dot] edu.

Solutions for 1-100

Int.	Representation	Alternative
1	5/5 – 5/5 + 5/5	5/5 + (5 – 5) + (5 – 5)
2	5/5 + 5/5 + (5 – 5)	(5 + 5)/5 + (5 – 5)/5
3	5/5 + 5/5 + 5/5	(5 + 5 + 5)/5 + (5 – 5)
4	(5 + 5)/5 + (5 + 5)/5	(5 + 5 + 5)/5 + 5/5
5	(5 + 5 + 5 + 5 + 5)/5	5*5 – 5 – 5 – 5 – 5
6	5 + 5/5 + (5 – 5)/5
7	5 + (5 + 5)/5 + (5 – 5)
8	5 + 5 – 5/5 – 5/5
9	5 + 5 – (5 + 5 – 5)/5
10	5 + 5 – 5/5 + 5/5
11	5 + 5 + (5 + 5 – 5)/5
12	5 + 5 + 5/5 + 5/5
13	5 + 5 + 5 – (5 + 5)/5
14	5*5 – 5 – 5 – 5/5
15	5 + 5 + 5 + (5 – 5)/5
16	5*5 – 5 – 5 + 5/5
17	5 + 5 + 5 + (5 + 5)/5
18	5*5 – 5 – (5 + 5)/5
19	5 + 5 + 5 + 5 – 5/5
20	5 + 5 + 5 + 5 + (5 – 5)
21	5 + 5 + 5 + 5 + 5/5
22	5*5 – (5 + 5 + 5)/5
23	5*5 – 5/5 – 5/5
24	5!/5 + 5/5 – 5/5
25	5*5 + 5*5 – 5*5
26	5*5 + 5/5 + (5 – 5)
27	5*5 + 5/5 + 5/5
28	5*5 + 5 – (5 + 5)/5
29	5!/5 + 5 + (5 – 5)/5
30	5 + 5 + 5 + 5 + 5 + 5
31	(5! + 5)/5 + 5 + 5/5
32	5*5 + 5 + (5 + 5)/5
33	(5! + 5!)/5 – 5 – 5 – 5
34	5*5 + 5 + 5 – 5/5
35	5*5 + 5 + 5 + (5 – 5)
36	(5! + 5 + 5)/5 + 5 + 5
37	(5! + 5! – 5)/5 – 5 – 5
38	5!/5 + 5!/5 – 5 – 5
39	5*5 + 5!/5 – 5 – 5
40	5*5 + 5*5 – 5 – 5
41	(5! + 5! – 5 – 5)/5 – 5
42	(5! + 5!)/5 – 5 – 5/5
43	(5! + 5!)/5 – 5 + (5 – 5)
44	(5! + 5!)/5 – 5 + 5/5
45	5*5 + 5 + 5 + 5 + 5
46	(5! – 5)/5 + (5! – 5)/5
47	5!/5 + 5!/5 – 5/5
48	5!/5 + 5!/5 + (5 – 5)
49	5*5 + 5*5 – 5/5	5!/5 + 5!/5 + 5/5
50	5*5 + 5*5 + (5 – 5)
51	5*5 + 5*5 + 5/5
52	5*(5 + 5) + (5 + 5)/5
53	5! – (5! + 5! + 5!)/5 + 5
54	5*(5 + 5) + 5 – 5/5
55	5*(5 + 5) + 5 + (5 – 5)
56	5*(5 + 5) + 5 + 5/5
57	(5! + 5! – 5)/5 + 5 + 5
58	5!/5 + 5!/5 + 5 + 5
59	5!/5 + 5*5 + 5 + 5
60	5*5 + 5*5 + 5 + 5
61	5*5!/(5 + 5) + 5/5
62	(5! + 5! + 5!)/5 – 5 – 5
63	(5! + 5!)/5 + 5 + 5 + 5
64	(5!/5!!)*(5!*5!!) + (5 – 5)	55 + 5 + 5 – 5/5
65	5*(5 + 5) + 5 + 5 + 5
66	5! – 5!/5 – 5*5 – 5
67	5! – 5!/5 – 5!/5 – 5
68	(5! + 5!)/5 + 5*5 – 5
69	5! – 5*(5 + 5) – 5/5
70	5*(5 + 5) + 5*5 – 5
71	(5! + 5! + 5!)/5 – 5/5
72	5!/5 + 5!/5 + 5!/5
73	(5! + 5! + 5!)/5 + 5/5
74	5*(5 + 5 + 5) – 5/5
75	5*5 + 5*5 + 5*5
76	5*(5 + 5 + 5) + 5/5
77	5! – 5!/5 – 5!/5 + 5
78	(5! + 5!)/5 + 5*5 + 5
79	5*(5 + 5) + 5 + 5!/5
80	5*(5 + 5) + 5*5 + 5
81	5! – 5!/5 – 5 – 5 – 5
82	(5! + 5! + 5!)/5 + 5 + 5
83	5! – 5*5 – 5!/(5 + 5)
84	5*5!/(5 + 5) + 5!/5
85	5*(5 + 5 + 5) + 5 + 5
86	(5*5! – 5!)/5 – 5 – 5
87	5! – (5! – 5)/5 – 5 – 5
88	(5!/5!!)*(5!*5!!) + 5!/5	5! + (5! – 5)/5 – 55
89	5! – 5*5 – 5 – 5/5
90	5! – 5*5 – 5 + (5 – 5)
91	5! – 5*5 – 5 + 5/5
92	5! – 5!/5 – 5 + 5/5
93	5! – 5*5 – (5 + 5)/5
94	5! – 5!/5 – (5 + 5)/5
95	5*5*5 – 5*5 – 5	5! – 5 – 5 – 5 – 5 – 5
96	5! – (5! – 5)/5 – 5/5
97	5! – 5*5 + (5 + 5)/5
98	5! – (5! – 5)/5 + 5/5
99	5! – 5*5 + 5 – 5/5
100	5*(5 + 5) + 5*(5 + 5)	5! – 5*5 + 5 + (5 – 5)

Solutions for 101-150

Int.	Representation	Alternative
101	5! – 5*5 + 5 + 5/5
102	5*5*5 – (5! – 5)/5
103	5! + 5 – (5! – 5 – 5)/5
104	5! – 5 – 5 – 5 – 5/5
105	5! – 5 – 5 – 5 + (5 – 5)
106	5! – 5 – 5 – 5 + 5/5
107	5! – (5! – 5)/5 + 5 + 5
108	5! – 5 – 5 – (5 + 5)/5
109	5! – 5 – (5*5 + 5)/5
110	5*(5*5 – 5) + 5 + 5
111	5! – 5!/5 + 5 + 5 + 5
112	5! – 5 – (5 + 5 + 5)/5
113	5! – (5*5 + 5 + 5)/5
114	5! – 5 – (5 + 5 – 5)/5
115	5! – 5 + (5 – 5) + (5 – 5)
116	5! – (5 + 5 + 5 + 5)/5
117	5! – 5/5 – (5 + 5)/5
118	5! – (5 + 5 + 5 – 5)/5
119	5! + 5/5 – (5 + 5)/5
120	5! + (5 + 5 – 5 – 5)/5
121	5! + 5/5 + (5 – 5)/5
122	5! + (5 + 5)/5 + (5 – 5)
123	5*5*5 – (5 + 5)/5
124	5! + (5 + 5 + 5 + 5)/5
125	5! + 5 + (5 – 5) + (5 – 5)
126	5! + 5 + 5/5 + (5 – 5)
127	5! + 5 + 5/5 + 5/5	5*5*5 + (5 + 5)/5
128	5! + 5 + (5 + 5 + 5)/5
129	5*5*5 + 5 – 5/5
130	5*5*5 + 5 + (5 – 5)
131	5*5*5 + 5 + 5/5
132	5! + 5!/(5 + 5) + (5 – 5)
133	5! + 5!/(5 + 5) + 5/5
134	5! + 5 + 5 + 5 – 5/5
135	5! + 5 + 5 + 5 + (5 – 5)
136	5! + 5 + 5 + 5 + 5/5
137	5! + (5! – 5 – 5)/5 – 5
138	5! + 5!/5 – 5 – 5/5
139	5! + 5*5 – 5 – 5/5
140	5! + 5*5 – 5 + (5 – 5)
141	5! + 5*5 – 5 + 5/5
142	5! + 5!/5 – (5 + 5)/5
143	5! + (5! – 5)/5 + (5 – 5)
144	5*5*5 + 5!/5 – 5
145	5*5*5 + 5*5 – 5
146	5! + 5!/5 + (5 + 5)/5
147	5! + (5! – 5 – 5)/5 + 5
148	5! + 5!/5 + 5 – 5/5
149	5! + 5*5 + 5 – 5/5
150	5! + 5*5 + 5 + (5 – 5)

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