Problems & Puzzles: Puzzles

Puzzle 611. Puzzle 2-3-5-7. Stephen Johnson and me constructed the following puzzle, which is a variation of some others constructed before out there of the same nature: C1: Using at the most four of these decimal digits: 2, 3, 5 & 7 where repetitions are allowed & counted. C2: Using any of these arithmetical six symbols: +, -, *, ^, ( & ) where repetitions are allowed & counted. Q. Get all the numbers from 0 to X in its minimal expression (minimal quantity Q of digits & arithmetical symbols) and stop whenever you can not get X+1 following the conditions C1 & C2. Valid examples: Follow the rules C1 & C2 and Q= minimal. 21=3*7, Q=3 13=3+3+5, Q=5 22=22, Q=2 24=3*7+3, Q=5 Invalid examples: Q is not minimal &/or fail at using C1 &/or C2. 13=22-3*3, Q=6, (Not minimal) 5=2+2+2/2, Q=7, (Not minimal & using an invalid arithmetical symbol) 24=2+3+5+7+7, Q=9, (Not minimal & using more than four prime integer) 4=4 Q=1, (Using an invalid integer)

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Contributions came from Giovanni Resta, Claudio Meller, Hakan Summakoğlu.

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All of them discovered that the first integer that can not be produced according to the rules given, is 179 and sent their respective expressions for the integers from 0 to 178.

It happened that the total sum of the symbols used in these integers from 0 to 178 is 774.

In their respective lists sent by the first time, Giovanni & Hakan used 775 symbols, while Claudio used 805 & Carlos Rivera used 809 symbols.

There are equivalent expressions for some integers that use the same quantity of symbols. For example: 20 = 23-3 = 22-2

This is one list using 174 symbols

n	Q	Expression
0	3	2-2
1	3	3-2
2	1	2
3	1	3
4	3	2+2
5	1	5
6	3	3+3
7	1	7
8	3	2^3
9	3	3^2
10	3	2*5
11	5	2^3+3
12	3	5+7
13	5	35-22
14	3	7+7
15	3	3*5
16	4	23-7
17	4	22-5
18	4	23-5
19	4	22-3
20	4	23-3
21	3	3*7
22	2	22
23	2	23
24	4	2+22
25	2	25
26	4	33-7
27	2	27
28	4	33-5
29	4	22+7
30	4	32-3
31	4	33-2
32	2	32
33	2	33
34	4	32+2
35	2	35
36	4	33+3
37	2	37
38	4	33+5
39	4	32+7
40	4	33+7
41	5	73-32
42	4	37+5
43	5	75-32
44	4	2*22
45	4	52-7
46	4	2*23
47	4	52-5
48	4	55-7
49	3	7*7
50	4	2*25
51	4	53-2
52	2	52
53	2	53
54	4	2*27
55	2	55
56	4	3+53
57	2	57
58	4	3+55
59	4	2+57
60	4	5+55
61	6	22*3-5
62	4	5+57
63	5	3*3*7
64	4	2*32
65	4	72-5
66	4	2*33
67	4	72-5
68	4	73-5
69	4	23*3
70	4	2*35
71	4	73-2
72	2	72
73	2	73
74	4	2*37
75	2	75
76	4	73+3
77	2	77
78	4	73+5
79	4	77+2
80	4	77+3
81	4	3*27
82	4	77+5
83	6	3+7+73
84	4	7+77
85	5	33+52
86	5	33+53
87	5	55+32
88	5	33+55
89	5	57+32
90	5	35+55
91	6	2*7+77
92	5	35+57
93	6	2^7-35
94	5	22+72
95	5	22+73
96	4	3*32
97	5	25+72
98	5	2*7*7
99	4	3*33
100	5	25+75
101	6	2^7-27
102	5	27+75
103	6	3*32+7
104	4	2*52
105	4	3*35
106	4	2*53
107	5	55+52
108	5	55+53
109	5	57+52
110	4	2*55
111	4	3*37
112	5	55+57
113	6	2+3*37
114	4	2*57
115	4	23*5
116	6	3*37+5
117	6	2*55+7
118	5	5^3-7
119	6	5+2*57
120	5	5^3-5
121	5	2^7-7
122	5	5^3-3
123	5	2^7-5
124	5	72+52
125	3	5^3
126	5	2^7-2
127	5	5^3+2
128	3	2^7
129	5	77+52
130	5	5^3+5
131	5	2^7+3
132	5	5^3+7
133	5	2^7+5
134	5	77+67
135	4	5*27
136	7	2^7+2^3
137	6	2+27*5
138	6	2*3*23
139	6	2*73-7
140	6	2*2*35
141	6	2*72-3
142	6	27*5+7
143	6	2*73-3
144	4	2*72
145	5	72+73
146	4	2*73
147	5	3*7^2
148	5	73+75
149	5	77+72
150	4	2*75
151	6	2^7+23
152	5	75+77
153	6	2^7+25
154	4	2*77
155	6	5+2*75
156	4	3*52
157	6	32+5^3
158	6	33+5^3
159	4	3*53
160	4	32*5
161	4	7*23
162	6	2+32*5
163	6	2^7+35
164	6	5+3*53
165	4	55*3
166	6	3^5-77
167	6	2+33*5
168	6	3+33*5
169	6	3*57-2
170	6	3^5-73
171	4	3*57
172	6	7+5*33
173	6	2+3*57
174	6	3*57+3
175	4	5*35
176	6	5+3*57
177	6	5^3+52
178	6	3+35*5

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