Now posted: The holeyest base

Given a positive integer as input, indicate which base from 2 to 16 gives the most holes in the representation of the input in that base.

The digits used are 0123456789ABCDEF. Note that these include upper case letters (the number of holes would be different for lower case letters).

Different fonts have different numbers of holes per character, so the number of holes per digit is as follows for this challenge (in the format digit : number of holes):

0 : 1
1 : 0
2 : 0
3 : 0
4 : 1
5 : 0
6 : 1
7 : 0
8 : 2
9 : 1
A : 1
B : 2
C : 0
D : 1
E : 0
F : 0

Input

• A positive integer

Output

• An integer from 2 to 16
• The output indicates which base gives the representation with the most holes (being the sum of the number of holes in each digit when represented in that base)
• For the purposes of counting holes:
  • Each representation will have no leading zeroes
  • The base indicator characters used by some languages, such as a leading 0b or 0x are not included in the counting
• If more than one base has the highest number of holes, any such base is a valid output

Example

For input 123 the representation in each of the bases from 2 to 16, and the associated number of holes, is as follows (in the format base : representation : holes):

2 : 1111011 : 0 + 0 + 0 + 0 + 1 + 0 + 0 = 1
3 : 11120 : 0 + 0 + 0 + 0 + 1 = 1
4 : 1323 : 0 + 0 + 0 + 0 = 0
5 : 443 : 1 + 1 + 0 = 2
6 : 323 : 0 + 0 + 0 = 0
7 : 234 : 0 + 0 + 1 = 1
8 : 173 : 0 + 0 + 0 = 0
9 : 146 : 0 + 1 + 1 = 2
10 : 123 : 0 + 0 + 0 = 0
11 : 102 : 0 + 1 + 0 = 1
12 : A3 : 1 + 0 = 1
13 : 96 : 1 + 1 = 2
14 : 8B : 2 + 2 = 4
15 : 83 : 2 + 0 = 2
16 : 7B : 0 + 2 = 2

The highest number of holes is 4, and this only occurs when the base is 14. So the only valid output is 14.

Test cases

Test cases with only 1 valid output

These test cases are in the format input : output.

2 : 2
3 : 3
4 : 2
7 : 7
8 : 2
10 : 2
12 : 2
16 : 2
17 : 2
18 : 2
20 : 2
24 : 2
27 : 3
32 : 2
33 : 2
34 : 2
35 : 2
36 : 2
37 : 2
38 : 2
40 : 2
41 : 2
42 : 2
48 : 2
49 : 2
50 : 2
51 : 2
54 : 3
59 : 12
60 : 13
61 : 13
62 : 9
63 : 13
64 : 2
65 : 2
66 : 2
67 : 2
68 : 2
69 : 2
70 : 2
72 : 2
73 : 2
74 : 2
76 : 2
77 : 2
80 : 2
82 : 2
84 : 2
85 : 2
87 : 3
94 : 11
95 : 14
96 : 2
97 : 2
98 : 2
123 : 14

Test cases including some with multiple valid outputs

These test cases are in the format input : [comma separated outputs] where any of the listed outputs would be valid. These also include all of the single valid output test cases listed above.

0 : [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
1 : [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
2 : [2]
3 : [3]
4 : [2]
5 : [2, 5]
6 : [2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
7 : [7]
8 : [2]
9 : [2, 3]
10 : [2]
11 : [12, 13, 14, 15, 16]
12 : [2]
13 : [2, 7, 9, 13, 14, 15, 16]
14 : [2, 5, 7, 8, 10, 14]
15 : [3, 5, 9, 11, 15]
16 : [2]
17 : [2]
18 : [2]
19 : [2, 11]
20 : [2]
21 : [2, 13]
22 : [2, 14]
23 : [12, 15]
24 : [2]
25 : [2, 5, 14]
26 : [2, 9, 15]
27 : [3]
28 : [2, 3, 6, 7, 10]
29 : [3, 5]
30 : [3, 11]
31 : [3, 7, 9, 11]
32 : [2]
33 : [2]
34 : [2]
35 : [2]
36 : [2]
37 : [2]
38 : [2]
39 : [2, 14]
40 : [2]
41 : [2]
42 : [2]
43 : [2, 16]
44 : [2, 9]
45 : [2, 3, 5]
46 : [2, 7, 10]
47 : [12, 13]
48 : [2]
49 : [2]
50 : [2]
51 : [2]
52 : [2, 11]
53 : [2, 7, 9, 11, 14, 15]
54 : [3]
55 : [3, 7]
56 : [2, 12]
57 : [2, 3, 12]
58 : [2, 9, 10, 12, 13]
59 : [12]
60 : [13]
61 : [13]
62 : [9]
63 : [13]
64 : [2]
65 : [2]
66 : [2]
67 : [2]
68 : [2]
69 : [2]
70 : [2]
71 : [2, 15]
72 : [2]
73 : [2]
74 : [2]
75 : [2, 16]
76 : [2]
77 : [2]
78 : [2, 9]
79 : [2, 5, 9]
80 : [2]
81 : [2, 3]
82 : [2]
83 : [2, 3, 12]
84 : [2]
85 : [2]
86 : [2, 10, 13]
87 : [3]
88 : [2, 10]
89 : [2, 9, 10, 13]
90 : [2, 3]
91 : [2, 3, 7, 11, 16]
92 : [2, 11, 14]
93 : [2, 3, 11, 14]
94 : [11]
95 : [14]
96 : [2]
97 : [2]
98 : [2]
99 : [2, 3]
123 : [14]
1200 : [2]
1201 : [2]
1202 : [2]
1203 : [2]
1204 : [2]
1205 : [2]
1206 : [2]
1207 : [2, 11]
1208 : [2]
1209 : [2]
1210 : [2]
1211 : [12, 16]
1212 : [2]
1213 : [2, 16]
1214 : [2, 9]
1215 : [3]
1216 : [2]
1217 : [2]
1218 : [2]
1219 : [2]
1220 : [2]
1221 : [2]
1222 : [2]
1223 : [2]
1224 : [2]
1225 : [2]
1226 : [2]
1227 : [2]
1228 : [2]
1229 : [2]
1230 : [2]
1231 : [2]
1232 : [2]
1233 : [2]
1234 : [2]
1235 : [2, 12]
1236 : [2]
1237 : [2]
1238 : [2]
1239 : [2]
1240 : [2]
1241 : [2]
1242 : [2]
1243 : [2, 14, 16]
1244 : [2]
1245 : [2, 5]
1246 : [2]
1247 : [12]
1248 : [2]
1249 : [2]
1250 : [2]
1251 : [2]
1252 : [2]
1253 : [2]
1254 : [2, 12]
1255 : [2, 12]
1256 : [2, 12]
1257 : [2, 12]
1258 : [2, 12]
1259 : [12]
1260 : [2]
1261 : [2]
1262 : [2, 11]
1263 : [2, 11, 12]
1264 : [2]
1265 : [2]
1266 : [2]
1267 : [2]
1268 : [2, 12]
1269 : [2, 3, 12]
1270 : [2, 5, 12]
1271 : [12]
1272 : [2]
1273 : [2]
1274 : [2, 5]
1275 : [2, 3, 5, 9, 12, 16]
1276 : [2, 12]
1277 : [9]
1278 : [12]
1279 : [5, 12]
1280 : [2]
1281 : [2]
1282 : [2]
1283 : [2]
1284 : [2]
1285 : [2]
1286 : [2]
1287 : [2]
1288 : [2]
1289 : [2]
1290 : [2]
1291 : [2]
1292 : [2]
1293 : [2]
1294 : [2]
1295 : [12]
1296 : [2]
1297 : [2]
1298 : [2]
1299 : [2]

Explanations in answers are optional, but I'm more likely to upvote answers that have one.

posted about 2 years ago

CC BY-SA 4.0

2y ago

trichoplax‭

1050 reputation 83 17 132 471