Timeline for Self-indulgent numbers
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 7, 2021 at 2:10 | comment | added | Gareth McCaughan♦ | (Note that that's 12345679, with the 8 missing, so it doesn't quite match up with your discovery.) It's fairly well known and I first encountered it long enough ago that I don't remember where. It's related to the fact that 1/(1-x)^2 = 1 + 2x + 3x^2 + ..., just as the fact that 1/9 = 0.11111... is related to the fact that 1/(1-x) = 1 + x + x^2 + ... . (Put x = 1/10.) You might find it amusing to look at the pattern in the decimal expansion of 1/243, too; I forget exactly why it does what it does, but it's the same circle of ideas. | |
| Apr 7, 2021 at 0:10 | comment | added | loopy walt | Actually, this 1/81 thing is good enough for me! Goes quite a bit of way to make it all plausible. For example, for k = 8 we get an almost clean argument: 8/81 = 1/9-1/81 = 0.1111111 - 0.012345, thus each digit d is replaced with 9-d. But how did you find it? WIth the benefit of hindsight, yes, of course, (1/9)^2 = 0.1111111^2 = 0.012345 etc. but it would have taken me forever to think of that myself. | |
| Apr 6, 2021 at 22:34 | comment | added | Gareth McCaughan♦ | Oh, those are rather cute. Beyond the observation that 1/81 = 0.012345679 recurring I don't see anything that looks like a fancy mathematical reason why they work, but I could well be missing something. | |
| Apr 6, 2021 at 17:12 | comment | added | loopy walt | I'm not sure it goes much beyond "here are two examples" but I'll show you anyway: 1234567890 (works for 1,2,4,5,7,8,..,22,23,25,26) and to a lesser extent 9876543210 (works for 1,2,4,5,7,8,10). I felt these are guessable enough to make an acceptable puzzle (for example, multiplying by 2 creates (viewed digit-by-digit) two copies of each even digit and 5 carries, just the right number to recreate the 5 odd digits, so trying permutations of 0123456789 is sort of educated guessing). I'd love to hear if you can see any deeper reason, also for the rather intriguing hit-hit-miss pattern. | |
| Apr 6, 2021 at 14:17 | comment | added | Gareth McCaughan♦ | Does that mean you have a simpler lower-brow solution? If so, and if it goes beyond "here is one example", I'd be interested to see it. | |
| Apr 6, 2021 at 1:22 | vote | accept | loopy walt | ||
| Apr 6, 2021 at 1:21 | comment | added | loopy walt | Wow, I hate to admit it but I wasn't aware that there is such a smart answer to my humble question! | |
| Apr 6, 2021 at 1:03 | history | answered | Gareth McCaughan♦ | CC BY-SA 4.0 |