# Talk:Sum to 100

## A 'sum' that can be shown but not expressed ?

Not sure that I have understood your third request. Something that can be 'shown' but not 'expressed' ? Sounds appeallingly ineffable and mysterious, but perhaps not entirely clear ...

Judging by your undo of my first suggestion, I guess that you are **not** simply looking for the first or lowest-valued integer than can't be expressed as a sum constructed in this way, but then I don't actually have any idea of what you **are** asking for. Perhaps some kind of example or concretisation might clarify ? Hout (talk) 01:42, 1 January 2017 (UTC)

- See the REXX's output to help understand what is required. I have changed the word
**first**to**lowest**which is much clearer to what was intended (I knew what I meant, but it was somewhat unclear in this context). (Note that negative sums/integers are possible, these sums are just negative "versions" of the positive sums, so the requirement was worded to avoid looking for negative results.) Since the sums are integers (as there is no way to get a non-integer sum from additions and/or subtractions of decimal digits or whole numbers). I used the word "expressed" as it isn't necessary to have the expression "shown". The word "shown" (to me) seems to indicate to display, not necessarily to express (in this case, mathematically). To answer your question, yes, it is possible. Showing something doesn't mean that it is being expressed (in the sense that it is being understood). I believe it boils down to word choices. I was trying for a concise wording of the requirement without being unnecessarily wordy (or obtuse). It's a fine art. -- Gerard Schildberger (talk) 02:04, 1 January 2017 (UTC)

- I've also added a trailing phrase to the 3
^{rd}requirement that (I think) fits in more to what you originally addressed. -- Gerard Schildberger (talk) 02:09, 1 January 2017 (UTC)

- Would it be consistent with your meaning to replace the word 'sum' with 'integer' ? Also if you are looking for the lowest integer that is inexpressible in these terms, perhaps it would be helpful to give an example of a higher integer of this kind, and demonstrate/explain its inexpressibility within this system ? Hout (talk) 02:16, 1 January 2017 (UTC)

- I used the word
**sum**because it was already used in the task's preamble and the word was also defined (in context). Introducing another word (integer) would, I think, just belabor the effort to have another version of the**sum**word. The next number (after**211**) that can't be expressed within the rules is**219**. Another example of a sum that can't be expressed is**5791**. I'm not sure how to demonstrate/explain its inexpressibility other than to show (display)*all*the possible mathematical expressions that**don't**(can't) express the**5791**sum. This would be somewhat of a strange proof (and voluminous), but not unheard of. -- Gerard Schildberger (talk) 02:29, 1 January 2017 (UTC)

- I used the word

- FWIW my experience as a reader is that using the word 'sum' at that point certainly proved confusing at first reading, and even now feels distractingly odd. 211 is certainly a
**number**– a member of the set of integers – but the point you are making about it seems to be precisely that it is**not**a sum (the result of an addition) within this system. Hout (talk) 02:38, 1 January 2017 (UTC)

- FWIW my experience as a reader is that using the word 'sum' at that point certainly proved confusing at first reading, and even now feels distractingly odd. 211 is certainly a

- I'm not saying that
**211**isn't an integer or a number --- just that it (the sum of 211) can't be expressed mathematically in the context of the rules for this puzzle as stated in the task's preamble.**211**is a sum (in particular, the lowest sum) of the set of sums that can't be expressed within the rules. -- Gerard Schildberger (talk) 02:56, 1 January 2017 (UTC)

- I'm not saying that

- PS on the issue of an example; an existence proof would be fine – you could just give an example of a number like 5791, say that it can't be expressed by a sum that is restricted to these permutations of digits and signs, and ask for code that finds the lowest positive integer that shares this property. Hout (talk) 02:46, 1 January 2017 (UTC)

- That seems overly wordy just to ask for the lowest positive sum that can't be expressed (within the rules). In particular, the requirement was expressly worded to find
*that*sum. However, I will add an example (i.e.: such as 5074). -- Gerard Schildberger (talk) 02:56, 1 January 2017 (UTC)

- That seems overly wordy just to ask for the lowest positive sum that can't be expressed (within the rules). In particular, the requirement was expressly worded to find

## a discussion of the task's requirements and wording

This (first) section of comments was moved here to the *talk* page from the task's requirements. -- Gerard Schildberger (talk) 18:07, 11 July 2017 (UTC)

Note: If you allow/disallow negative first numbers you do get different results. If you allow negative first numbers, which you should according to the specification; then you get at least two targets with the same highest number of solutions: The positive and negative versions of the same target, only the positive one is between 0 to 123456789. The specification assumes there is only one positive target with the maximum number of solutions. It is worth looking for multiple targets with the same number of solutions, just in case there are more. This is especially true if you try the variants with different lists of digits. For example using the digits '987654321' and excluding a negative first number comes up with two distinct targets with the same maximum number of solutions, both of which happen to be positive. If you are playing with the asymetric version with out leading negative numbers of different lists of digits the sums with the highest number of solution may only be negative. In this case it is worth excluding the test to show only positive totals and put up with seeing duplicate +ve and -ve solutions.

Robin Murison

- Added signature of Robin Murison by Gerard Schildberger (talk) 18:07, 11 July 2017 (UTC)

## extra credit

I had initially thought to add an extra credit solution for the *largest* sum (except for the obvious **123456789**). I'm not sure it would be a worthwhile (optional) requirement, it could be pretty CPU intensive. -- Gerard Schildberger (talk) 04:21, 1 January 2017 (UTC)

- Seems fine as an additional task – I've just added it for Haskell and it doesn't seem to noticeably increase the interpreter run-time. Hout (talk) 00:29, 5 January 2017 (UTC)

- For interpreted languages, it may be problematic. Even so, I added an "extra credit" (optional) requirement. -- Gerard Schildberger (talk) 00:44, 5 January 2017 (UTC)

- I might not be optimistic with AppleScript, but JavaScript and GHCi (the Haskell interpreter that comes with the GHC compiler) both seem quite relaxed about it when run from inside a code editor. Of course the data types and algorithm will make a difference :-) Hout (talk) 01:01, 5 January 2017 (UTC)

- For performance note that it is not necessary to generate the strings to count the solutions. There are 2 * 3 ** (base-2) possible strings. In base 2 the maximum value is 1 the minimum value -1. 0 can not be generated so 66% of values can be represented. The number of possible strings increases exponentially with the base, so to stretch this task for extra credit we could ask what is the %age in base 64, or 60 if we want to be Babylonian, or why not more. I often read that we use base 10 because we have 10 fingers, is there any archeological evidence that the Babylonians were severely deformed? --Nigel Galloway (talk) 12:54, 25 February 2017 (UTC)
- I know you're kidding, and of course the Babylonians had ten fingers, but the assumption is that they realized base 60 would be an advantageous base and chose it with intention, rather than just going with the number of digits they were born with. I don't know of anyone else who independently adopted base 60, but base 10 was far from universal around the world; the Mayans used base 20, the Hawaiians base 12, some other Pacific Islanders base 4... --Markjreed (talk) 00:54, 25 March 2024 (UTC)
- Apparently there
*is*indeed some archeological evidence that nearly all Babylonians had*twelve*finger knuckles on the back of each hand, as well as*five*digits on the other side. And, albeit less impressively, ancient Mayan skeletons show that both of their hands (also with five digits apiece) were also connected to the rest of them by so-called "wrists"... (IKUAKAT) --Petelomax (talk) 04:42, 25 March 2024 (UTC)

- For performance note that it is not necessary to generate the strings to count the solutions. There are 2 * 3 ** (base-2) possible strings. In base 2 the maximum value is 1 the minimum value -1. 0 can not be generated so 66% of values can be represented. The number of possible strings increases exponentially with the base, so to stretch this task for extra credit we could ask what is the %age in base 64, or 60 if we want to be Babylonian, or why not more. I often read that we use base 10 because we have 10 fingers, is there any archeological evidence that the Babylonians were severely deformed? --Nigel Galloway (talk) 12:54, 25 February 2017 (UTC)

## complete list of solutions for **9**

For those that are interested (or need a sanity check), the 46 solutions (for the sum of 9) are:

solution: 9 ◄───► -1+2+3+4+5+6+7-8-9 solution: 9 ◄───► -1+2+3-4+5-6-7+8+9 solution: 9 ◄───► -1+2+3-4-5+6+7-8+9 solution: 9 ◄───► -1+2+3-45+67-8-9 solution: 9 ◄───► -1+2+34+56+7-89 solution: 9 ◄───► -1+2-3+4+5-6+7-8+9 solution: 9 ◄───► -1+2-3+4-5+6+7+8-9 solution: 9 ◄───► -1+23+4+5+67-89 solution: 9 ◄───► -1+23+4-5-6-7-8+9 solution: 9 ◄───► -1+23-4+5-6-7+8-9 solution: 9 ◄───► -1+23-4-5+6+7-8-9 solution: 9 ◄───► -1-2+3+4+5+6-7-8+9 solution: 9 ◄───► -1-2+3+4+5-6+7+8-9 solution: 9 ◄───► -1-2+3-4-5-6+7+8+9 solution: 9 ◄───► -1-2+3-4-56+78-9 solution: 9 ◄───► -1-2-3+4-5+6-7+8+9 solution: 9 ◄───► -1-2-3+45-6-7-8-9 solution: 9 ◄───► -1-2-3-4+5+6+7-8+9 solution: 9 ◄───► -1-2-34+56+7-8-9 solution: 9 ◄───► -1-23+45-6-7-8+9 solution: 9 ◄───► -12+3-45-6+78-9 solution: 9 ◄───► -12+34+5+6-7-8-9 solution: 9 ◄───► -12+34+56-78+9 solution: 9 ◄───► -12-34+5+67-8-9 solution: 9 ◄───► 1+2+3+4-5-6-7+8+9 solution: 9 ◄───► 1+2+3-4+5-6+7-8+9 solution: 9 ◄───► 1+2+3-4-5+6+7+8-9 solution: 9 ◄───► 1+2-3+4+5+6-7-8+9 solution: 9 ◄───► 1+2-3+4+5-6+7+8-9 solution: 9 ◄───► 1+2-3-4-5-6+7+8+9 solution: 9 ◄───► 1+2-3-4-56+78-9 solution: 9 ◄───► 1+2-34-56+7+89 solution: 9 ◄───► 1+23+4-5-6-7+8-9 solution: 9 ◄───► 1+23-4+5-6+7-8-9 solution: 9 ◄───► 1+23-45+6+7+8+9 solution: 9 ◄───► 1-2+3+4+5+6-7+8-9 solution: 9 ◄───► 1-2+3-4-5+6-7+8+9 solution: 9 ◄───► 1-2-3+4+5-6-7+8+9 solution: 9 ◄───► 1-2-3+4-5+6+7-8+9 solution: 9 ◄───► 1-2-3-4+5+6+7+8-9 solution: 9 ◄───► 1-2-3-4-5-67+89 solution: 9 ◄───► 1-23+4+5-67+89 solution: 9 ◄───► 1-23+45-6-7+8-9 solution: 9 ◄───► 1-23-4+5+6+7+8+9 solution: 9 ◄───► 1-23-45-6-7+89 solution: 9 ◄───► 12-34-56+78+9 46 solutions found for 9

## complete list of the number of solutions for zero ──► 300

For those that are interested in the density of solutions:

22 solutions found for 0 ────────────────────── 43 solutions found for 1 ─────────────────────────────────────────── 18 solutions found for 2 ────────────────── 41 solutions found for 3 ───────────────────────────────────────── 18 solutions found for 4 ────────────────── 40 solutions found for 5 ──────────────────────────────────────── 24 solutions found for 6 ──────────────────────── 39 solutions found for 7 ─────────────────────────────────────── 18 solutions found for 8 ────────────────── 46 solutions found for 9 ────────────────────────────────────────────── 17 solutions found for 10 ───────────────── 38 solutions found for 11 ────────────────────────────────────── 27 solutions found for 12 ─────────────────────────── 38 solutions found for 13 ────────────────────────────────────── 24 solutions found for 14 ──────────────────────── 43 solutions found for 15 ─────────────────────────────────────────── 18 solutions found for 16 ────────────────── 39 solutions found for 17 ─────────────────────────────────────── 23 solutions found for 18 ─────────────────────── 37 solutions found for 19 ───────────────────────────────────── 23 solutions found for 20 ─────────────────────── 43 solutions found for 21 ─────────────────────────────────────────── 25 solutions found for 22 ───────────────────────── 36 solutions found for 23 ──────────────────────────────────── 32 solutions found for 24 ──────────────────────────────── 36 solutions found for 25 ──────────────────────────────────── 25 solutions found for 26 ───────────────────────── 44 solutions found for 27 ──────────────────────────────────────────── 25 solutions found for 28 ───────────────────────── 35 solutions found for 29 ─────────────────────────────────── 34 solutions found for 30 ────────────────────────────────── 31 solutions found for 31 ─────────────────────────────── 26 solutions found for 32 ────────────────────────── 37 solutions found for 33 ───────────────────────────────────── 24 solutions found for 34 ──────────────────────── 35 solutions found for 35 ─────────────────────────────────── 32 solutions found for 36 ──────────────────────────────── 32 solutions found for 37 ──────────────────────────────── 27 solutions found for 38 ─────────────────────────── 37 solutions found for 39 ───────────────────────────────────── 31 solutions found for 40 ─────────────────────────────── 26 solutions found for 41 ────────────────────────── 34 solutions found for 42 ────────────────────────────────── 34 solutions found for 43 ────────────────────────────────── 29 solutions found for 44 ───────────────────────────── 42 solutions found for 45 ────────────────────────────────────────── 27 solutions found for 46 ─────────────────────────── 27 solutions found for 47 ─────────────────────────── 35 solutions found for 48 ─────────────────────────────────── 26 solutions found for 49 ────────────────────────── 28 solutions found for 50 ──────────────────────────── 35 solutions found for 51 ─────────────────────────────────── 29 solutions found for 52 ───────────────────────────── 33 solutions found for 53 ───────────────────────────────── 30 solutions found for 54 ────────────────────────────── 26 solutions found for 55 ────────────────────────── 23 solutions found for 56 ─────────────────────── 29 solutions found for 57 ───────────────────────────── 32 solutions found for 58 ──────────────────────────────── 25 solutions found for 59 ───────────────────────── 33 solutions found for 60 ───────────────────────────────── 30 solutions found for 61 ────────────────────────────── 24 solutions found for 62 ──────────────────────── 34 solutions found for 63 ────────────────────────────────── 22 solutions found for 64 ────────────────────── 26 solutions found for 65 ────────────────────────── 32 solutions found for 66 ──────────────────────────────── 30 solutions found for 67 ────────────────────────────── 25 solutions found for 68 ───────────────────────── 29 solutions found for 69 ───────────────────────────── 22 solutions found for 70 ────────────────────── 25 solutions found for 71 ───────────────────────── 26 solutions found for 72 ────────────────────────── 24 solutions found for 73 ──────────────────────── 22 solutions found for 74 ────────────────────── 26 solutions found for 75 ────────────────────────── 25 solutions found for 76 ───────────────────────── 22 solutions found for 77 ────────────────────── 24 solutions found for 78 ──────────────────────── 25 solutions found for 79 ───────────────────────── 19 solutions found for 80 ─────────────────── 31 solutions found for 81 ─────────────────────────────── 20 solutions found for 82 ──────────────────── 23 solutions found for 83 ─────────────────────── 25 solutions found for 84 ───────────────────────── 21 solutions found for 85 ───────────────────── 18 solutions found for 86 ────────────────── 23 solutions found for 87 ─────────────────────── 17 solutions found for 88 ───────────────── 24 solutions found for 89 ──────────────────────── 21 solutions found for 90 ───────────────────── 22 solutions found for 91 ────────────────────── 21 solutions found for 92 ───────────────────── 23 solutions found for 93 ─────────────────────── 17 solutions found for 94 ───────────────── 17 solutions found for 95 ───────────────── 19 solutions found for 96 ─────────────────── 21 solutions found for 97 ───────────────────── 15 solutions found for 98 ─────────────── 25 solutions found for 99 ───────────────────────── 12 solutions found for 100 ──────────── 18 solutions found for 101 ────────────────── 19 solutions found for 102 ─────────────────── 14 solutions found for 103 ────────────── 14 solutions found for 104 ────────────── 18 solutions found for 105 ────────────────── 18 solutions found for 106 ────────────────── 17 solutions found for 107 ───────────────── 19 solutions found for 108 ─────────────────── 12 solutions found for 109 ──────────── 14 solutions found for 110 ────────────── 19 solutions found for 111 ─────────────────── 13 solutions found for 112 ───────────── 12 solutions found for 113 ──────────── 17 solutions found for 114 ───────────────── 15 solutions found for 115 ─────────────── 14 solutions found for 116 ────────────── 20 solutions found for 117 ──────────────────── 13 solutions found for 118 ───────────── 11 solutions found for 119 ─────────── 18 solutions found for 120 ────────────────── 10 solutions found for 121 ────────── 9 solutions found for 122 ───────── 13 solutions found for 123 ───────────── 12 solutions found for 124 ──────────── 12 solutions found for 125 ──────────── 19 solutions found for 126 ─────────────────── 8 solutions found for 127 ──────── 12 solutions found for 128 ──────────── 12 solutions found for 129 ──────────── 13 solutions found for 130 ───────────── 7 solutions found for 131 ─────── 15 solutions found for 132 ─────────────── 14 solutions found for 133 ────────────── 8 solutions found for 134 ──────── 16 solutions found for 135 ──────────────── 10 solutions found for 136 ────────── 5 solutions found for 137 ───── 15 solutions found for 138 ─────────────── 7 solutions found for 139 ─────── 10 solutions found for 140 ────────── 10 solutions found for 141 ────────── 10 solutions found for 142 ────────── 9 solutions found for 143 ───────── 14 solutions found for 144 ────────────── 8 solutions found for 145 ──────── 8 solutions found for 146 ──────── 11 solutions found for 147 ─────────── 9 solutions found for 148 ───────── 6 solutions found for 149 ────── 10 solutions found for 150 ────────── 9 solutions found for 151 ───────── 6 solutions found for 152 ────── 13 solutions found for 153 ───────────── 9 solutions found for 154 ───────── 6 solutions found for 155 ────── 10 solutions found for 156 ────────── 8 solutions found for 157 ──────── 4 solutions found for 158 ──── 8 solutions found for 159 ──────── 5 solutions found for 160 ───── 4 solutions found for 161 ──── 11 solutions found for 162 ─────────── 5 solutions found for 163 ───── 6 solutions found for 164 ────── 7 solutions found for 165 ─────── 4 solutions found for 166 ──── 5 solutions found for 167 ───── 3 solutions found for 168 ─── 8 solutions found for 169 ──────── 4 solutions found for 170 ──── 10 solutions found for 171 ────────── 6 solutions found for 172 ────── 4 solutions found for 173 ──── 5 solutions found for 174 ───── 3 solutions found for 175 ─── 1 solution found for 176 ─ 6 solutions found for 177 ────── 3 solutions found for 178 ─── 3 solutions found for 179 ─── 7 solutions found for 180 ─────── 2 solutions found for 181 ── 4 solutions found for 182 ──── 4 solutions found for 183 ──── 3 solutions found for 184 ─── 5 solutions found for 185 ───── 4 solutions found for 186 ──── 5 solutions found for 187 ───── 4 solutions found for 188 ──── 6 solutions found for 189 ────── 6 solutions found for 190 ────── 1 solution found for 191 ─ 4 solutions found for 192 ──── 2 solutions found for 193 ── 1 solution found for 194 ─ 4 solutions found for 195 ──── 1 solution found for 196 ─ 3 solutions found for 197 ─── 7 solutions found for 198 ─────── 3 solutions found for 199 ─── 3 solutions found for 200 ─── 4 solutions found for 201 ──── 2 solutions found for 202 ── 3 solutions found for 203 ─── 2 solutions found for 204 ── 2 solutions found for 205 ── 2 solutions found for 206 ── 6 solutions found for 207 ────── 4 solutions found for 208 ──── 1 solution found for 209 ─ 5 solutions found for 210 ───── no solutions found for 211 4 solutions found for 212 ──── 1 solution found for 213 ─ 3 solutions found for 214 ─── 1 solution found for 215 ─ 6 solutions found for 216 ────── 1 solution found for 217 ─ 3 solutions found for 218 ─── no solutions found for 219 3 solutions found for 220 ─── no solutions found for 221 4 solutions found for 222 ──── 1 solution found for 223 ─ 4 solutions found for 224 ──── 2 solutions found for 225 ── 5 solutions found for 226 ───── no solutions found for 227 4 solutions found for 228 ──── no solutions found for 229 3 solutions found for 230 ─── 2 solutions found for 231 ── 2 solutions found for 232 ── no solutions found for 233 5 solutions found for 234 ───── no solutions found for 235 3 solutions found for 236 ─── 1 solution found for 237 ─ 3 solutions found for 238 ─── no solutions found for 239 6 solutions found for 240 ────── no solutions found for 241 5 solutions found for 242 ───── 3 solutions found for 243 ─── 6 solutions found for 244 ────── 2 solutions found for 245 ── 5 solutions found for 246 ───── no solutions found for 247 2 solutions found for 248 ── 3 solutions found for 249 ─── 2 solutions found for 250 ── 2 solutions found for 251 ── 7 solutions found for 252 ─────── 1 solution found for 253 ─ 4 solutions found for 254 ──── 2 solutions found for 255 ── 5 solutions found for 256 ───── 2 solutions found for 257 ── 7 solutions found for 258 ─────── 1 solution found for 259 ─ 4 solutions found for 260 ──── 3 solutions found for 261 ─── 3 solutions found for 262 ─── 1 solution found for 263 ─ 4 solutions found for 264 ──── 4 solutions found for 265 ──── 2 solutions found for 266 ── 5 solutions found for 267 ───── 2 solutions found for 268 ── 2 solutions found for 269 ── 6 solutions found for 270 ────── 2 solutions found for 271 ── 1 solution found for 272 ─ 3 solutions found for 273 ─── 1 solution found for 274 ─ 1 solution found for 275 ─ 3 solutions found for 276 ─── no solutions found for 277 4 solutions found for 278 ──── 3 solutions found for 279 ─── 4 solutions found for 280 ──── 4 solutions found for 281 ──── 2 solutions found for 282 ── 4 solutions found for 283 ──── no solutions found for 284 2 solutions found for 285 ── no solutions found for 286 no solutions found for 287 2 solutions found for 288 ── 1 solution found for 289 ─ 1 solution found for 290 ─ 2 solutions found for 291 ── 1 solution found for 292 ─ 1 solution found for 293 ─ 3 solutions found for 294 ─── 1 solution found for 295 ─ 3 solutions found for 296 ─── 2 solutions found for 297 ── 2 solutions found for 298 ── 2 solutions found for 299 ── no solutions found for 300 sum of 9 has the maximum number of solutions: 46 sums of 211 219 221 227 229 233 235 239 241 247 277 284 286 287 300 have the minimum number of solutions: 0

## a discussion of the task's requirements and wording

This (added) section of superscript comments was moved here to the *talk* page from the task's requirements (the task's preamble). I also changed the hard-to-read superscript text to be displayed as "regular" text). -- Gerard Schildberger (talk) 18:12, 11 July 2017 (UTC)

- Note: If you allow/disallow negative first numbers you do get different results. If you allow negative first numbers, which you should according to the specification; then you get at least two targets with the same highest number of solutions: The positive and negative versions of the same target, only the positive one is between 0 to 123456789. The specification assumes there is only one positive target with the maximum number of solutions. It is worth looking for multiple targets with the same number of solutions, just in case there are more. This is especially true if you try the variants with different lists of digits. For example using the digits '987654321' and excluding a negative first number comes up with two distinct targets with the same maximum number of solutions, both of which happen to be positive. If you are playing with the asymetric version with out leading negative numbers of different lists of digits the sums with the highest number of solution may only be negative. In this case it is worth excluding the test to show only positive totals and put up with seeing duplicate +ve and -ve solutions.

- Robin Murison

- Added signature of
*Robin Murison*by Gerard Schildberger (talk) 18:12, 11 July 2017 (UTC)

- Added signature of

Yes, a minus sign (**-**) is allowed before the first digit(s) This is specifically allowed from the 2^{nd} sentence in the task's preamble, there is no **if** about it.

There is only one sum (you call it a target) that has a maximum. This wasn't assumed, it was determined by examination of all possible targets (which were non-negative sums), in part, to give programmers *a priori* information. The 2^{nd} task requirement asks **the** sum which has the maximum number of solutions (from zero to infinity --- this excludes negative sums/targets). There is only one correct answer --- but you may assume there will be more than one solution, and if so, the wording will be changed. The task's preamble could've been worded as asking for **sum(s)**; it was a word choice that I made when composing the task's requirements.

Note that negative sums (targets) are specifically excluded as per the task's description/requirements.

It *is* worth looking for multiple targets (and indeed, that is a requirement of this task) with the same --- or for that matter --- any number of solutions, this is a purpose of this task.

The use of the digit string of **987654321** isn't allowed, only the digit string of **123456789**.

-- Gerard Schildberger (talk) 19:07, 11 July 2017 (UTC)