7 ms·
Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)
- sp332 10y agoWhy did this paper need four revisions? Edit: the revisions have different ranges of numbers, but they go up and down. Odd.
- dotancohen 10y agoActually, four is even.
- deleted 10y ago[deleted]
- MylesHiIdebrand 10y agoWhy is this even a thing?
- jlarocco 10y agoThe abstract and introduction don't explain this very well. My understanding is that the author wrote the digits 1 through 9 in ascending order, and then inserted parenthesis, addition, subtraction, multiplication, division and exponentiation operators between them where appropriate to get every number from 0 to 11111. And then he did the same thing using the digits 9 through 1 in descending order and did the same thing. He was able to find a solution for every number except 10958 with 1-9 in ascending order. EDIT: The paper also doesn't explain much about how or why they did this. At first glance I don't think it would be too difficult to come up with a branch and bound algorithm to search through all the solutions.
- pavel_lishin 10y agoI wonder if any numbers have multiple solutions.
- mmanfrin 10y agoI am nearly certain many do.
- Beltiras 10y agoYour sibling comment has an example in 1+2+3=1* 2*3
- Cogito 10y agoAn easy way to try this is to consider any that start with 1+2+3. You can create a second version by replacing this with 1×2×3. For example, 10914 = 1 × 2 × 3 × (4^5 + 6 + 789) 10914 = (1 + 2 + 3) × (4^5 + 6 + 789)
- squidbot 10y agoI found this article about the paper and there are some comments from the author. Not much more enlightening that it was something of a "cocktail party game." https://nebusresearch.wordpress.com/2013/06/10/counting-from-52-to-11108/#more-1778 https://nebusresearch.wordpress.com/2013/06/10/counting-from...
- jwtadvice 10y agoAt my office we lost a lot of time because someone put up a challenge to represent all the numbers 0-99 with algebriac expressions containing exactly four digits '4'. The use of powers and radicals was allowed. I myself solved several numbers before exhausting the tricks I could think of, and reverted to brute force/search with Polish/prefix notation, then realizing how massive the search space was for even just a few operators. Adding parentheses ordering further explodes the search space. Finally, while this kind of concrete and explicit expression is "toy math", it is related to Godel Sentences and lexigraphic ordering of proofs used through much of the establishment of logic, the foundations of proof systems, and computer science.
- ninju 10y agoMaybe something neat to bring up as a "drinking game"...at my next Math Symposium Pick at random 3 digit number and try to express using the digits 1 thru 9...loser has to drink
- javajosh 10y agoIt wasn't clear why this was interesting from the title. But here's an example: 0 = 12 + 34 − 56 − 7 + 8 + 9. 1 = 1^23456789. 2 = 123+4−56−78+9. 3 = 123 − 45 − 6 − 78 + 9. And then in decreasing order 0 = 98−7−6−54−32+1. 1 = 98−76−54+32+1. 2 = 9+87−65+4−32−1. 3 = 98−76−5+4+3−21. That this is possible at all is counter-intuitive to me, and hence fascinating. Even more fascinating is that he does not have a solution for 10958 in the ascending case.
- happimess 10y agoIt's not clear from your formatting, but the line was 1 = 1 ^ 23456789 I thought it was really cool that this pattern is used elsewhere to collapse any prefix into 1, e.g. 9 = (1 ^ 2345678) * 9
- javajosh 10y agoThanks. Copy/pasted from PDF and it got lost. Fixed.
- deleted 10y ago[deleted]
- monk_e_boy 10y agoI didn't read the article... but why 1 to 9? Is it because we have 10 fingers and 10 symbols that represent numbers? We're this numbers chosen for another reason?
- andrepd 10y agoThere are 10 one-digit numbers in decimal. 0 is excluded because of obvious reasons, leaving numbers 1 through 9.
- monochromatic 10y agoIn other words, yes, we use decimal because we have ten fingers.
- monochromatic 10y agoThis is way more numerology than math, and I don't see any deep results coming out of this. Was this just a fun side project for the author?
- thomasahle 10y agoNot everything that's written as an article is serious :)
- daveguy 10y agoOne persons numerology is another person's abstract algebra. I think this was mostly just for fun/curiosity, but operation combinations are interesting -- if nothing else to show how easy numerology can be. It's not like the author attached significance to any of the calculations. Here is a result (we can call it Taneja's Conjecture): The digits 1-9 in sequence can not be combined with fundamental operations (+ - / * ^) in a way that computes the value 10958.
- hodgesrm 10y agoA lot of math results are curiosities at first but turn out to be important in unexpected ways. The original proof of the existence of irrational numbers depends on using the Pythagorean theorem to show that the leg of an isosceles right triangle is simultaneously odd and even if irrational numbers do not exist. The proof works because somebody had thought out the seemingly trivial notions of what it means for a number to be odd and even and the effect this has on their squares and ratios. [1] I have always had this picture of somebody's spouse in 500 BC yelling at him to get in and do some useful work instead of standing around in the square drawing numbers. [1] https://en.wikipedia.org/wiki/Irrational_number#Ancient_Greece https://en.wikipedia.org/wiki/Irrational_number#Ancient_Gree...
- aaron695 10y agoWould be more interested in the script than the list and how long it took to run. And if this is the best so far? Seems surprising given it's a popular kids game. It's not really maths IMO since I think using 98 is cheating for instance, it's really (9 X 10 + 8), 10 not being allowed. So kids are not learning real maths, more arithmetic.
- lanna 10y agoThere is no script, he did it mostly by hand.
- aaron695 10y agoI would have thought that to be the fun bit! I think there was at least one script for some of the numbers, this I would have thought the most interesting, optimisation of the script and for all numbers and proving some numbers have no solutions([edit] possibly very hard) - "The author is thankful to xxxx in finding some difficult representations using computerized script."
- lanna 10y ago"some". In previous versions of the paper, he claims to have done everything manually: "To bring these numbers, thousands of combinations are considered without use of any programming language."
- adtac 10y agoI just wrote a tiny script that does this: https://github.com/adtac/123456789/blob/master/main.py https://github.com/adtac/123456789/blob/master/main.py
- thomasahle 10y agoHuh, he does 98 as 98=1×23+45+6+7+8+9. Or do you mean that 23 and 45 are cheating, because they are two digit numbers? I think they rules are quite well explained and followed.
- adtac 10y agoI just wrote a script to do this in, like, 5 minutes :) And the whole thing runs in 7 seconds on my machine. https://github.com/adtac/123456789/blob/master/output https://github.com/adtac/123456789/blob/master/output Admittedly, it doesn't have every number. I suspect it's because I haven't included bracketed expressions.
- 7373737373 10y agoWhat happens if you allow for 0?
- metaphor 10y agoFor ascending solution taking a lazy approach, it appears to be a trivial inclusion, considering for any n with a solution using 1-9, a solution which includes 0 will simply take the form: n = 0 + <ascending_solution_n> without any additional consideration. Similarly for descending solution: n = <descending_solution_n> + 0 The descending case also looks like it'll produce a crap ton of solutions for n = 1 of the form: 1 = <arbitrary_descending_permutation> ^ 0
- 7373737373 10y agoMultiplication makes a difference here
- metaphor 10y ago3 = 123+4+5-6*7^89 First verifiable instance of error? Did a little fishing for overflow candidates with the following regex: [1-9]{3,}\^[1-9]{3,} ...caught 58 results between 266 and 10940: 266 = 123^456-78-9 ... 10940 = 12345^6789 My kung fu isn't Python so I don't know how types are handled. That eval() function sure smells suicidal for anything remotely serious though.
- witty_username 10y agoNo, Python doesn't have integer overflow. "^" is the XOR operator. GP's code has that bug. > That eval() function sure smells suicidal for anything remotely serious though. Yes, ast.literal_eval() should be used.
- chriswarbo 10y agoSeems related to this recent Numberphile video which does a similar thing for e and pi: https://www.youtube.com/watch?v=xgBGibfLD-U https://www.youtube.com/watch?v=xgBGibfLD-U
- Grue3 10y agoInteresting fact: every integer can be written with three 2's.
- rerx 10y agon + (2 - 2) / 2 ?
- Grue3 10y agoNope. You can use three instances of number 2, all arithmetical operations and elementary special functions that are taught in high school.
- lifthrasiir 10y ago-log_2 log_2 sqrt(...(sqrt(2))...) {with n copies of sqrt} = -log_2 (2^-n log_2 2) = -log_2 (2^-n) = -(-n) = n. I would say that you need only one 2 if you replace log_2 by lg, so we should make lg taught in high schools... wink
- iplaw 10y agoSo, who is going to find the increasing solution for 10958?