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Challenges

# The holeyest base

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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

• A single 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, but you must only output one of them

## 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 potential outputs] where any one of the listed outputs would be valid. These also include all of the single valid output test cases listed above.

Note that the output can be any one, but must be only one, of the potential outputs listed.

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.

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# Japt-h, 19 18 bytes

Gõ ñ@ìX xg#5#}4s3


Try it

Gõ ñ@ìX xg#5#}4s3     :Implicit input of integer U
G                      :16
õ                     :Range [1,G]
ñ                   :Sort by
@                  :Passing each X through the following function
ìX                :  Convert U to base X digit array
x              :  Reduce by addition after
g             :    Indexing into
#5#}4       :      14551254
s3     :      Converted to base-3 string
:Implicit output of last element


"But, Shaggy, why ... no, how is that 14551254?" "Glad you asked! See this tip here"

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## Sclipting, (UTF-16) 86 bytes

標갠 관上가①要❶❹剩걉눑감⓶重右갰雙⓷加⓶❸分終⓷棄終并❶訂乾⓶折❷同終長괐縮⒈棄丟


Explanation:

Input n

갠 관上 for b from 2 to 16
가 s = 0
① n' = n
要 while n' is not 0
❶❹剩 x = n' mod b
걉눑감 byte array 00 00 01 00 10 01 01 10 00 01 00 01 00 00 00 01
⓶重右 shift right by 2x
갰雙 bitwise & with 0b11
⓶❸分 n' = n' // b
終
⓷棄

❶訂乾 Get the max value
⓶折❷同終長괐縮⒈棄丟 Get the index of the max value

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# Vyxal, 19 18 bytes

16Ḣλ»≤ṪḢ»3τ?nτİ∑;∴


Try it Online!

The link goes to a program which has every test case.

-1 thanks to Steffan

## Explained (old)

16Ḣλ?$τ»≤ṪḢ»3τ$İ∑;O
16Ḣ                 # The range [2, 16]
λ             ;O # Get the maximal items in the list by:
∑   #   Summing
?$τ # The input in that base$İ    #   Indexed into (as in, vectorised index)
»≤ṪḢ»3τ      #   ⟨ 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 0 | 0 ⟩ (14551254 converted to base 3 - this is a 0-indexed list representing the number of holes in each letter)

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18 bytes: 16Ḣλ»≤ṪḢ»3τ?nτİ∑;∴ (1 comment)
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# Jelly, 20 bytes

³b‘ị“ḃṘ¤’b3¤S
⁴ḊÇÐṀḢ


Try it online!

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# Ruby, 63 bytes

->n{(2..16).max_by{|x|n.digits(x).sum{45088003.digits(3)[_1]}}}


Attempt This Online!

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# Ruby, 77 bytes

->x{(2..16).max_by{x.to_s(_1).chars.sum{|y|"".bytes[y.hex]}}}


Attempt This Online!

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