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Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

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Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#51

Post by daveguy »

monochromatic wrote: post #18

This 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?

One 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.

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#52

Post by dastbe »

Phemist wrote: post #39
adtac wrote: post #23

I 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 Admittedly, it doesn't have every number. I suspect it's because I haven't included bracketed expressions.

I think a good idea might be to do it for postfix/prefix notation, rather than infix. This will also get rid of the need for parentheses.

The ordering of numbers will be different between infix, prefix, and postfix representations.

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#53

Post by hodgesrm »

monochromatic wrote: post #18

This 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?

A 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_Gree...

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#54

Post by javajosh »

Retra wrote: post #21
javajosh wrote: post #10

It 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 fo…

There's not a lot interesting about "facts about numbers" when you place such a low upper bound on your achievements. If he'd only done this up to 100, he'd never have had trouble finding a solution for 10958. But why 11111? Why not 111111111111? To me this is just as interesting as any other arbitrary list of permutations of symbols.

Intuitively, this problem seems isomorphic to the famous NP-complete problem, 3SAT[1]. Instead of picking binary states, you have to pick symbols to insert, and where, that yield the desired outcome. If this intuition is true, then if you can write a program to generate these expressions (in polynomial time) then you'd have also proven P=NP. You could probably brute force it though as the difficulty scales with the length of your target number.

1. https://en.wikipedia.org/wiki/Boolean_satisfiability_problem

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#55

Post by saghm »

nullc wrote: post #28
saghm wrote: post #26

Earlier quoted context omitted.

Is that actually why base 10 became the norm? I've never really learned why exactly decimal was chosen, and I'd be interested in learning about what historical evidence there is for its origin.

It's often claimed, but I'm not aware of any strong evidence for it. In particular, since our hands can show "10" the natural base this would imply is base _11_ and not 10. Another argument I've heard presented is that since 10 is divisible by 2 and 5 this simplifies many common calculations. There were groups in the past advocating use of other bases (in particular 12 but also 16). The mayan's number system is base…

> The mayan's number system is base 20 (though the digits are are drawn out of sub digits in base-5).

I think I remember learning in school that one of the ancient Mesopotamian civilizations (Sumerian, I think, but I could be wrong) used 60 as the basis for their system because it was divisible by 1, 2, 3, 4, 5, and 6.

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#56

Post by 7373737373 »

metaphor wrote: post #33
7373737373 wrote: post #25

Earlier quoted context omitted.

What happens if you allow for 0?

For 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 + without any additional consideration. Similarly for 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 = ^ 0

Multiplication makes a difference here

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#57

Post by squidbot »

jlarocco wrote: post #5

The 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 th…

I 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...

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#58

Post by questerzen »

adtac wrote: post #23

I 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 Admittedly, it doesn't have every number. I suspect it's because I haven't included bracketed expressions.

I added division and brackets. This gets every number up to 5397, with only about 400 numbers still missing overall (Potentiation is a problem though (particularly when combined with brackets) as Python becomes impossibly slow using bignums. (I estimate it will take several days of compute time to test all 250 million possibilities, unless I put some effort into speeding it up significantly.) But at least it would prove definitively whether 10958 is possible.

If the only goal were to search for a solution to 10958 though, it should be fairly straightforward to re-order the search in approximately ascending order of compute cost so that if a simple solution exists it will be found quickly.

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#59

Post by andrepd »

nullc wrote: post #28
saghm wrote: post #26

Earlier quoted context omitted.

Is that actually why base 10 became the norm? I've never really learned why exactly decimal was chosen, and I'd be interested in learning about what historical evidence there is for its origin.

It's often claimed, but I'm not aware of any strong evidence for it. In particular, since our hands can show "10" the natural base this would imply is base _11_ and not 10. Another argument I've heard presented is that since 10 is divisible by 2 and 5 this simplifies many common calculations. There were groups in the past advocating use of other bases (in particular 12 but also 16). The mayan's number system is base…

11? How so? If I have 10 fingers I can count from 1 to 10. When I start again I count from 11 to 20. Again from 21 to 30. In base 11 it would go 1 to A, 10 to 19, 1A to 28, etc. It doesn't make sense.

Re: Numbers 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 (2014)

#60

Post by saghm »

throwaway982749 wrote: post #27
saghm wrote: post #26

Earlier quoted context omitted.

Is that actually why base 10 became the norm? I've never really learned why exactly decimal was chosen, and I'd be interested in learning about what historical evidence there is for its origin.

Well... there's evidence right there in the English language. dig·it [ˈdijit] NOUN 1) any of the numerals from 0 to 9, especially when forming part of a number. 2) a finger (including the thumb) or toe.

Right, but that doesn't really say anything about causation; there is definitely a linguistic link between the body parts and numerical digits in this word, but that could have easily developed after base ten became common for other reasons.
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