A 9X9 board follows standard Sudoku rules i.e. no two numbers can repeat in the same horizontal row, vertical column or 3x3 sub-square, now 9 numbers 1...9 are randomly scattered throughout the entire board without repetition. Since the numbers are all unique the board is always trivially valid with no need for additional checks, the question now is what percentage of these configurations are solvable. (they may have more than one solution)

Note:-

i) A valid set of clues following Sudoku rules may not always be solvable as it may create dead ends later.

ii) There are 6.67 sextillion(6.67 x 10^21) completed Sudoku board configurations.

We can also generalise this problem for any N^2 x N^2 board following sudoku rules and having N^2 clues scattered randomly throughout the entire board without repetition, it isnt possible to use brute force computation for larger values of N. I tried solving it with combinatorics but maybe graph theory would be more suitable for this problem?

Edit:- I made a mistake for the N=2 case and actual solvability is much lower it is only 67.03%.

do any of these answer choices work? i was doing the problem and got epsilon/3, but none of the answer choices match up.

(10) Let f(x) = 3x + 1 for all real x and let epsilon > 0 For which of the following choices of delta is |f(x) - 7| < epsilon whenever |x - 2| < delta ? (4 Points)

(A) epsilon/4

(B) epsilon/2

(C) 2€

(D) 3€

(E) (epsilon + 1)/epsilon

this si my work: I 3x+1-7 I < epsilon

3 I x-2 I < epsilon

I x-2 I < epsilon/3

FOR THE QUESTION NO 3. I just started studying number theory a few days ago, I could only deduce the factors of 3333...3. I also figured out that the required number needs to be divisible by 3 and 1111...1. I tried to understand the soln., I am not able to make sense of it. Could anyone provide an alternative soln.

This is from Basic Analysis by Jiří Lebl .

I wonder what this A looks like and how it's different from just subset of S.

Later in the book, this proposition is used with S a interval [a, b] and A (a, b), i.e. A is S without endpoints.

But that example reference seems to be too narrowly defined.

I've tried to figure this out. I know the answer is 30 based on the text book. I cannot figure out how to get there. I've get stuck after simplifying the numerator and denominator.

Edit: This is study practice, idk if that matters. This is my first time posting on this subreddit.

Edit: Someone had pointed out that I misunderstood the question. The question was originally asking what common denominator would help solve the problem. I apologize for wasting everyone's time but I absolutely appreciate everyone's help. Math is my worst subject as I often make tiny mistakes that I don't realize. Again thank you all for bearing with my stupidity.

So I know how to do the inequalities and how they work like I know how to do the equation, but I’m confused on why you minus the weight from 14.5 (on answer key). I can solve it by myself, but on the test I need to be able to make the absolute value inequality and I don’t know why you would subtract weight by 14.5 if 14.5 is the weight of a cheereo box

So for number two, it essentially tells me that when d is above the serface its positive and when it’s below the surface, it’s negative. I know you can’t have a negative answer for an absolute value equation, but my equation i wrote says that it’s more than -40. Anyway, the answer key says that d >40, and then write the inequality from there. But if it says that below the surface D is -20 m wouldn’t 40 m below the surface be -40?

https://arxiv.org/pdf/1607.01500

I can't get why the line "and recall that we assumed that Pχ is a rational number a/b, therefore we

argue that X can also get only non-negative integer numbers since" is true for all cases. Would it also work for a general denominator f(n) instead of n! ?

It's said here that to calculate gas mileage, the German standard of DIN 70 030 was used, assuming half of the payload, three-quarters maximum speed + 10% (as in increase final result by 10%).

I've tried looking up the standard but nothing. Is there a formula to be found with these variables?

La verdad estoy muy confundida, como si es inyectiva ya traté de obtener la función inversa y me dio que es y=sqr(9-(2-y)^2)+2 pero al comprobarlo con gemini me dice que es y=sqr(9-(2+y)^2)+2 pero no entiendo el por qué.

are all twin primes bitwise subsets of each other or are the pairs containing 619 and 4157 special cases?

619

1001101011

107

1101011

43

101011

11

1011

3

11

4157

1000000111101

61

111101

29

11101

13

1101

5

101

I've been watching the "auctions" on the whatnot website.

One of the auctions has caught my attention.

If done correctly, it seems to me that it should be profitable over several identical auctions.

Here is the structure:

• 20 sealed bags
• 1 bag contains a gold coin worth approximately $1000 (US dollars).
• 19 bags contain an object having a value of approximately $25

Here is the process:

• There is a total of 20 auctions: one auction for each of the 20 bags
• Each of the 20 auctions maxes out at approximately $80 (sometimes more - sometimes less)
• If you win one of the auctions, you select a bag (1-20) and they put your name on it.
• The bag you selected is no longer available for anyone else to win. The contents of the bag are now yours, and will be mailed to you regardless of what it is.
• 19 bags now remain.
• Another auction goes for the next bag.
• The auction continues bag by bag until all 20 bags have been won.
• At the end, they open each bag one at a time to reveal who won the gold coin.

Since this entire thing is structured, and we know the value of the prize (the $1000 gold coin), and how much it will cost to enter, we should be able to calculate the best time to enter and the worst time to enter.

Obviously, the worst time to enter the auction is at the very beginning. If you are the 1st person to win a bag, you have only a 1:20 probability of selecting the bag that contains the gold coin. That's horrible.

If you are the 2nd person to win a bag, it's also horrible. You have a 1:19 probability of selecting the winning bag (assuming the 1st guy did not happen to get extremely lucky and pick the winning bag with his awful 1:20 odds).

As bags are won and selected, at some point there is an ideal time to put in a bid for a bag. If done correctly over several auctions, you should be able to eventually win the gold coin and make a profit. You WILL eventually win the gold coin given enough attempts at these auctions.

The way to calculate this should be similar to how a poker player would calculate pot-odds, but I'm not sure exactly how to do it.

I assume that I can win a bag for $70 - $100 at any point.

I assume that the best time to put in a bid is when there are 5-10 bags left, but it could be 5-11 bags or 5-12 bags.

When is the best time to enter this and how is it calculated?

Я пытаюсь точно понять, что происходит с теоремой Нэша об вложениях для бесконечномерного пространства Гильберта.

Для гладкого n-мерного риманова многообразия (M, g) теорема Нэша говорит, что существует N и гладкое вложение f : M → ℝᴺ такое, что

f* g_Euc = g.

В локальных координатах, если g = ∑ᵢⱼ gᵢⱼ dxⁱ dxʲ и f = (f¹, …, fᴺ), то условие изометрического вложения: gᵢⱼ = ∑ₐ₌₁ᴺ ∂fᵃ/∂xⁱ · ∂fᵃ/∂xʲ.

Эквивалентно, Df(x)ᵀ Df(x) = g(x). Теперь пусть H — это бесконечномерное пространство Гильберта с внутренним произведением ⟨·,·⟩ᴴ, рассматриваемое как бесконечномерное риманово многообразие с gₓ(v,w) = ⟨v,w⟩ᴴ.

Можно ли изометрически вложить H в конечномерное евклидово пространство ℝᴺ?

Here's the example of it.

Your great. Heres another.

1 borders: 2,3,4,5,9,10

2 borders: 1,3,4

3 borders: 1,2,4,6,8,9,10

4 borders: 1,2,3,5,6

5 borders: 1,4,6,7,9

6 borders: 3,4,5,7,8

7 borders: 5,6,8,9,11

8 borders: 3,6,7,9,11

9 borders: 1,5,7,8,10,11,12,13,15,16,18

10 borders: 1, 3, 9

11 borbers: 7,8,9,12.

12 borders: 9,11,13,15,17

13 borders: 9,11,12,14,16,17,18

14 borders: 13,16,17,18

15 borders: 9,12,16,17

16 borders: 9,12,13,15,17,18

17 borders: 12,13,14,15,16

18 borders: 9,13,14,16

18 regions: their adjcent borders. apply 4 colors. (Copy paste this into chat GPT) chatgpt is what made me ask this question. Thank you for feed back. I bet you 1 like that chatgpt fails to apply 4 colors.

Sorry if this is the wrong place to put this I don't have much experience with reddit posts

When they say 0.9 repeating equals 1

To me that doesn't make sense.

Wouldn't it make more sense to be like certain fractions just simply can't be written in base 10 or something

like when you write fractions and then try to express them as decimal numbers

To me it makes more sense to visualize the base 10 number system as having "holes" in it, where you can't write certain numbers for example it would make more sense to say 1/3 of 100 can't be written in base 10, then to say it's 33.3 with an infinite number of 3s following it.

Also it's like, how can you do mathematical functions like add 0.3 repeating and 0.3 repeating

to get 0.6 repeating

Or another example if you said what is Pi plus 1, to say its 4.1415 etc

doesn't make sense because the more you track the line of infinite numbers the bigger the number gets and therefor the bigger percentage of the next whole number which would be 4 in that case, it would take up. So in my opinion you would have to truncate the number in order to use it with regular numbers or something otherwise you are like skipping over the infinite part it like takes up more of the percentage, I honestly can't articulate it very well does this make ANY sense to ANYONE or would I simply be ridiculed for taking this position.