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... keep reading on reddit β‘I want to figure out how to represent any given natural number with only the number 1, addition, multiplication, and parentheses, with the fewest number of 1s. I have an algorithm that I think does a good job, but I don't know if it's provably the best for each case.
For each number:
So, for example, for 59:
This is 14 1s. Will doing it this way always produce the best answer for any natural number?
https://www.greentechmedia.com/articles/read/winter-storm-forces-blackouts-across-texas
"Unprecedented winter storms have hit Texas so hardΒ and forced so many power plants offlineΒ that rolling grid outages that began Sunday night have grown into hours-long blackouts leaving about 2.5 million customers without power on Monday morning. Until those power plants can be brought back online, itβs unclear how many people will be left powerless and for how long.
Thatβs the dire report from Texas grid operator ERCOTΒ after its supply-demand balance grew much worse than expected through Sunday evening and into Monday morning. The historic storm that spurred a statewideΒ emergency declarationΒ from Gov. Greg Abbott on Friday has forced power plants and wind farms to stop generating electricity, triggering statewide emergency blackouts and leaving the timeline for restoring that power unclear.
Texas has become the epicenter in a series of winter storms that has blanketed the central U.S. and spurred storm watches in 40 states.Β ERCOT hit a record winter peak demand of about 69.2Β gigawatts on Sunday evening.Β Demand for electricity to heat and power homes, businesses and factories remained high through the evening and into Monday morning, Dan Woodfin, ERCOTβs senior director of system operations, said in a Monday press conference.
At the same time, a number of natural gas, coal and nuclear thermal generators began tripping offline starting around 1:30 a.m. Monday, he said. While ERCOT hasnβt yet collected the data to determine the precise causes of those generator outages, a previous report fromΒ a 2014 cold snapΒ suggests a range of causes, fromΒ natural-gas pipelines freezing upΒ to the failure of equipment that's needed to keep power plants operating safely"
Summary: Texas and most states that don't generally deal with cold weather are massively unprepared and all grid systems must be updated to higher standards to handle load surges and renewable energy. This effected ALL energy sources.
I'm interested to find out, for each natural number (particularly just 1-100), the way or ways to represent the number with just the number 1, addition, multiplication, and parentheses, using the fewest number of 1s.
As an example, I can do several short ones by hand:
4 = (1+1)Γ(1+1), or 1+1+1+1
6 = (1+1+1)Γ(1+1)
This should be similar to the Countdown numbers game, of which there are many solvers out there, but I can't find an existing solver that will A) let me enter more than 6 starting numbers (every starting number in this case would be 1), B) let me constrain the operations to just + and Γ, and C) output the guaranteed shortest answer. There are solvers that tick some of those requirements, but I can't find one that will do all three.
It might be easier to output the result in postfix notation:
4 = 1 1 1 1 + + +
6 = 1 1 1 + + 1 1 + Γ
(These are the kind of questions I have that make me wish I could program, but every time I try to start learning, the whole thing seems so huge and daunting, like I need to have so much back-knowledge before I can even start to craft something I really want to make.)
Recently I learned that there is for 3 x 3 boards, Tic Tac Toe is solved with with a forced draw assuming best play from both players. I then started wondering if it held true for larger boards. If it isn't known, why not? If it is, how was it figured out?
Clarification: You need n in a row to win a game
Problem from Kansai University this year.
I hear a lot about how Euler's number and the natural logarithm are incredibly important things that describe the way the natural world behaves, yet I am falling short on fining any concrete examples of it outside of pure mathematics and obviously compound interest which is most people's introduction to e.
By defination, natural numbers are all integers that are positive. Since zero is neither positive and nor negative, how is it being used in Bachet's conjuncture?
According to the conjuncture, any natural number X can be represented as a sum of squares of four other natural numbers,
X = Xa^{2} + Xb^{2} + Xc^{2} + Xd^{2}
And mostly, they use Zero in the equation. But zero isn't a natural number...
Am I missing something? I'd really like to know.
I was looking into hyperreal numbers out of sheer curiosity and the section on the construction of the hyperreals stated that the existence of ultrafilters on the natural numbers is guaranteed by Zorn's lemma but can't be explicitly constructed, but it provides no context as to why. I would appreciate if someone could explain the reason for that to me.
I have little to no background in the areas of math that I would need to understand this (I've yet to take any courses remotely close to abstract math save for the unit on Boolean algebra in highschool geomemtry), so an intuitive explanation would be preferred.
I studied Real Analysis before and using Peano Postulate , the natural number start with one ; but I am studying a lecture about Physics about topological spaces , before all of that , we use set theory to generate natural number from empty set, set empty set name as 0 . Therefore the natural number is 0.
Some people said depend one what you name you first number , it would include 0 or 1, it should not matter but it is an inconsistency I found it really annoying.
Those two things reminded me that there are some really fascinating things about the real world today. So fascinating that you would think that a really skilled worldbuilder would have come up with it first, had you not known it wasn't their idea, in fact, it existed IRL! So that made me ask myself...
Are there things that are so fascinating in real life that it made you think that, had you not known it existed in real life, you'd be thinking that it was the idea of a master worldbuilder? It can be anything.
The answer is 10 and 11, but I have no idea how you get there.
I made an spreadsheets which ranks all countries by:
The overall rank is a weighted formula from all above mentioned factors, where the total number of tourists is weighted the strongest (around 70%) and the GDP per is weighted with 25%.
All data are collected from the database of the worldbank.
Here is the link to the google spreadsheet:
https://docs.google.com/spreadsheets/d/16HdpW73wAaPjDr69o9vK1aoxkQ_EkzX1jDc_hOnM3wo/edit?usp=sharing
How I thought about it:
There are two cases: one where you have all 3 digits odd and one where you have 2 even digits and 1 odd digit.
First case: you have 3 odd digits, so all 3 are from this set {1, 3, 5, 7, 9}. You have 5^3 = 125 numbers for this case.
Second case: you have 2 even digits and an odd one.
Case 2. a. If first digit is odd then it still can't be 0, else it would be a 2-digit number. First digit can take values from this set {2, 4, 6, 8}. The other two need to be one from the set {1, 3, 5, 7, 9} and one from {0, 2, 4, 6, 8} so you have 4*5^2 numbers in this case.
Case 2. b. If first digit is odd then it can be any of the 5 odd digits, and you have the other two digits be any of the even digits so you get another 5^3 = 125 numbers
The total amount of numbers that satisfy the condition is 5^3 + 5^3 + 4*5^2 = 350
The total number of 3 digit numbers is 900 so the probability is 350/900 = 7/18
However, the answers section in my book says that the probability is 125/900 = 5/36 because "there are 5^3 = 125 favorable cases". Why?
Suppose I have an ordered field F and 1 β F. Then we define the set of natural numbers recursively, here we call it N_F in the field F via
1 β F_N, and if n β N_F then n*+1 β N_F, and* n = 1+ .... +1, where the right side has n (not bold n) summands.
I want to prove this recurrence relation satisfies the Peano axioms. These kinds of questions look so basic so I don't what tools should I use to prove it/... I just know N_F is defined as an element 1 and some successors to complete the set.
1.For proving 1 is not the successor of any k β N_F. I am thinking contradiction and showing that there is some k β N_F, and I have that k+1=1 (since its a successor). But I don't know what to state next, I know I cannot explicitly define 0 since it's not an element of N_F.
For proving if n,m β N_F, and m+1=n+1, then n=m, this also seems obvious for me but I don't know how to prove it rigorously, as I don't know how to get rid of the "1" on each side(maybe using the fact that F is an ordered field)
For S β N_F, I want to show that S=N_F if i) 1 β S and ii) if n β S, then n+1βS
Here I know that SβN_F is already assumed and I only have to prove that N_FβS, but its very obvious since S is defined exactly like N_F and I don't know how to put them in words.
Any inputs are appreciated don;t know where to start.
N = {0, 1, 2, 3, 4, 5 ...}
or
N = {1, 2, 3, 4, 5 ...}
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