7 ms·
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
by javajosh 10y ago
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 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.
- saghm 10y agoIs 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.
- throwaway982749 10y agoWell... 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.
- saghm 10y agoRight, 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.
- nullc 10y agoIt'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 20 (though the digits are are drawn out of sub digits in base-5).
- saghm 10y ago> Another argument I've heard presented is that since 10 is divisible by 2 and 5 this simplifies many common calculations. To me, it always seemed like 5 was only a common number because of its place in base 10, not the other way around. There doesn't seem to be any reason that 2 and 5 would have been chosen as more significant than 2 and 3 or 2 and 7.
- Retra 10y agoThere'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.
- matthewbauer 10y agoI assume 111112 is unrepresentable as well?
- Mizza 10y agoTo me, the pattern of unrepresentable numbers (like the pattern of primes) would also be an "interesting" result.
- javajosh 10y agoIntuitively, 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 https://en.wikipedia.org/wiki/Boolean_satisfiability_problem