Squares Definition and 400 Threads

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted



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{\displaystyle \square }
ABCD.

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  1. O

    A question about perfect squares

    Homework Statement I've been given a task to find "A 4-digit perfect square whose digits are all unique, and whose square root is a prime number". That's all. I know that there are about 10 possible answers and I need them all. Thanks a lot for any future help.
  2. K

    Why Use the Least Squares Method for Finding Slope?

    Homework Statement why did you use the least squares method for finding m, rather than the standard slope formula? Homework Equations The Attempt at a Solution I am totally confused about why you have to use the least squares method
  3. camilus

    Volume of an Open-Top Box with 3-Sided Squares

    Homework Statement An open-top box can be formed from a rectangular piece of cardboard by cutting equal squares from the four corners and then folding up the four sections that stick out. For a particular-sized piece of cardboard, the same volume results whether squares of side one or...
  4. D

    Limit Problem Difference of Squares

    how would you do: lim x->3+ of abs(x+3) / x^2 - 9 because if you take the difference of squares of the bottom and divide x+3 / x+3 for the positive abs case you are left with 1/(x-3) with x approaching 3 making the denominater 0 I have since realized that the question was asking for...
  5. M

    Solving Punnet Squares for Dark Hair & Brown Eyes

    [SOLVED] Punnet Squares The allele for dark hair D is dominant over that of red hair d, and the allele for brown eyes B is dominant over that of blue eyes b. A women is red-haired and has blue eyes. Her husband is dark-haired and has brown eyes. a) What are the possible genotypes for the...
  6. D

    Method of Least Squares Linear Fitting

    Homework Statement An experiment was conducted on a liquid at varying temperatures and the volume obtained at the differing temperatures are as follows: V/cm3 θ/oC 1.032 10 1.063 20 1.094 29.5 1.125 39.5 1.156 50 1.186 60.5 1.215 69.5 1.244 79.5 1.273 90 1.3 99 Assume that V...
  7. H

    Constrained Cubic Polynomial Fitting: A Bezier Approach?

    Anybody know the math/theory behind linear least squares where the curve is forced to go through the first and last data points?I'm specifically dealing with cubic polynomials. In standard linear least squares formulation (i.e. ATAc = ATy) the curve doesn't, in general, go through any of...
  8. Q

    Magic Squares & Cubes: Uncovering a 3D Arithmetic Sequence

    Hello to you all! I've been involved in Magic Squares & Cubes for the last 8-9 years. I've recently developed 3-D models of that, too. What I've observed is a unique relationship between simple Arithmetic Sequence & Magic Squares & Cubes. Combining the two, I've reached a new 3 Dimensional...
  9. Gib Z

    How can the minimum diagonal be used to solve 6c?

    [img=http://img527.imageshack.us/img527/9639/96652845hm1.th.jpg] I can do 6.a) and b) but still need to know how to do c). I've done half of the question because I have an example of one where the diagonal adds up to 33, but I can't prove that's the smallest diagonal sum. Dont know where...
  10. N

    Determine the remainders when dividing their squares by four

    Randomly select eight odd integers of less than 1000 a) Determine the remainders when dividing their squares by four, and tabualte your results b) Make a conjecture about your findings c) test your conjecture with at least five larger integers d) Prove of justify the conjecture you make. n...
  11. M

    Understanding the Difference of Squares in Limits: A Comprehensive Guide

    I'm reviewing material for my exams and I came across this: \lim _{x\rightarrow \infty }\sqrt {{x}^{2}+x+1}-\sqrt {{x}^{2}-3\,x} The only explanation it gives is "By the difference of squares" the solution sheet then jumps to: \lim _{x\rightarrow \infty }{\frac {4\,x+1}{\sqrt...
  12. M

    Least Squares Solution - Or is there?

    I have a problem that says to find the least squares solution to \newcommand{\colv}[2] {\left(\begin{array}{c} #1 \\ #2 \end{array}\right)} K x = \colv{2}{2} for K = \left( \begin{array}{cc} 1 & 2\\ 2 & 4 \end{array} \right). Then express the solution in the form x = w + z, where w is in the...
  13. I

    What is the least possible sum of squares when the sum of two numbers is 20?

    Homework Statement The sum of two numbers is 20. What is the least possible sum of their squares. 2. The attempt at a solution Before I show my work, I'm pretty sure I have the answer. I think it's 200. If you add 10 and 10, you will have 20. If you square 10 you get 100, thus the sum...
  14. F

    Can all primes of the form 4n + 1 be written as the sum of two squares?

    Homework Statement I must prove the theorem that if the GCD of a and b is 1, and if p is an odd prime which divides a^2 + b^2, p is of the form 4n + 1. Homework Equations I have seen two proofs that I think might be helpful. 1. If a and b are relatively prime then every factor of...
  15. B

    Minimizing Distance Between Two Lines

    Can someone help with the folowing? Suppose L1 is the line through the origin in the direction of a1 and L2 is the line through b in the direction of a2. I am supposed to find the closest points x1a1 and b+x2a2 on the two lines. So I am trying to find the equations that would minize...
  16. mattmns

    (Ugly?) Inequalities - Squares and sums

    Here is the question from the book: ------ Let n\geq1 and let a_1,...,a_n and b_1,...,b_n be real numbers. Verify the identity: \left(\sum_{i=1}^n{a_ib_i}\right)^2 + \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\left(a_ib_j-a_jb_i\right)^2}} =...
  17. B

    Find Perfect Squares from Residues Mod 16

    Homework Statement X mod m is the remainder when x is divided by m. This value is called a residue. Find all perfect squares from the set of residues mod 16. The Attempt at a Solution There was a suggestion that this would become clearer when the definition of perfect square was...
  18. X

    Is There a Square of a Rational Number Between Any Two Positive Rationals?

    Hi all, I think this sounds like a really simple and trivial question, but I've no clue as to where i should start: true or false? between any two different positive rational numbers lies the square of a rational number. while i can't provide a construction of such a number, i somehow...
  19. G

    Connection between cubed binomial and summation formula proof (for squares)

    I was reading through a proof of the summation formula for a sequence of consecutive squares (12 22 + 32 + ... + n2), and the beginning of the proof states that we should take the formula: (k+1)3 = k3 + 3k2 + 3k + 1 And take "k = 1,2,3,...,n-1, n" to get n formulas which can then be...
  20. S

    Minimize the sum of the squares

    Hello i need help with a question, other people tried to help me, i just cannot get it! its driving me crazy Two positive numbers have sum n. What is the smallest value possible for the sum of their squares? so i have n=x+y x>0 y>0 y=n-x we want to minimize S S=x^2+y^2...
  21. N

    Exploring the Pattern of Prime Numbers and Squares

    I am curious as to whether this pattern will always hold true: Let's say we take the prime numbers: 2,3,5,7,11,13,17,19,23...primes and we take the square(individually) minus 1 3,8,24,48,120,168,288,360,528...p^2 - 1 Then starting with the third p^2 - 1 (24), all of the p^2 - 1 can be...
  22. D

    Finding Pythagorean Triples: Sums of Two Squares

    Which squares are expressible as the sum of two squares? Is there a simple expression I can write down that will give me all of them? Some of them? Parametrization of the pythagorean triples doesn't seem to help.
  23. D

    Exploring the Cardinality of X: A Set of Squares

    Suppose X is a set consisting of squares with the property that any addition with elements of X (where no two are the same) gives a square (might not be in X). How many elements can X have?
  24. P

    Can Magic Squares be Applied in Mathematical and Scientific Research?

    Greetings, I'm curious if "magic squares" have been found to be useful in mathematical or scientific endeavors apart from an "oddity" or "game"
  25. R

    Composite numbers and squares from recursive series

    This has to do with the 2nd order recursive sequence \{...a, b, c ...\} where a,b,c are any three sucessive terms and c = 6b-a + 2k. I found that it has the following property. 8ab - (a+b-k)^2 = 8bc - (b+c-k) That is eight times the product of two adjacent terms always equals the square of...
  26. C

    Direct expression for sum of squares

    How do you go from \sum_{n = 1}^n i^2 to \frac{n(n + 1)(2n + 1)}{6}?
  27. C

    The least squares approximation - best fit lines revisited

    We all know the least squares method to find the best fit line for a collection of random data. But I wonder if it is the best method. Suppose we have two random variables y and x that appear to have a linear relation of the type y = ax+b. What we want is, given the next type x signal to...
  28. K

    Linear Algebra: Least Squares and vectors

    Hi, I was working on a problem and I can't figure out what I'm supposed to do. It reads, find the vector in subspace S that is closest to v; write v as the sum of a vector in S and a vector in S^a; and find the distance from v to S. S spanned by {(1,3,4)} v = (2,-5,1) Ok, what I did was...
  29. G

    Least squares and integration problem

    Question states Consider the vector space C[-1,1] with an inner product defined by <f,g> = the integral from 1 to -1 of f(x)g(x) dx a) Show that u1(x)= 1/(2^.5) u2(x)= ((6^.5)/2)x form an orthonormal set of vectors b) Use the result from a) to find the best least squates...
  30. S

    Finding Perfect Squares of n Factored as a^4*b^3*c^7

    a number n when factorised can be written as a^4*b^3*c^7.find number of perfect square which are factors of n.a,b,c are prime >2. I have no idea how to start? please help.
  31. V

    Dividing a 5-Square Cross into 4 Equal Squares

    A figure contains five equal squares in the form of a cross. Can you show how to divide this figure into four equal parts which will fit together to form a square
  32. H

    Linear Regression, Linear Least Squares, Least Squares, Non-linear Least Squares

    It seems to me that Linear Regression and Linear Least Squares are often used interchangeably, but I believe there to be subtle differences between the two. From what I can tell (for simplicity let's assume the uncertainity is in y only), Linear Regression refers to the general case of fitting...
  33. S

    How Do You Derive the Formula for the Sum of Consecutive Squares?

    ok, i need to derive a forumla that will add the consecutive squares of n numbers. for example 1^2 + 2^2 + 3^2 + ... + (n-2)^2 + (n-1)^2 + (n)^2 I have worked on this problem for quite some time and haven't been able to come up with anything. I do know that the sum of consecutive...
  34. M

    Odd Integer Squares: Proving 8k+1

    Prove that the square of an odd integer is always of the 8k + 1, where k is an integer. Any help would be appreciated.
  35. A

    Least squares regression problem

    Hi, I am having some difficulty with this problem: what would be Y^h^a^t if s_y_/_x = 439, n = 24 and 95% confidence interval estimate for the average Y given a particular value of X is 1125 and 1695. ----------------- I know Y^h^a^t = b_o + b_1x but I am not sure how I can use the...
  36. M

    How Do You Calculate Variance Between Combined Data Sets?

    I have a statistics test coming up and we were given two really hard problems to figure out. I don't quite know what they are asking, and we are kinda on our own to solve it. Any help would be greatly appreciated. 1. Given that the equation for the sum of the squares is SS = x2 - (x)2/n...
  37. J

    Covering the 62 Squares with Dominoes

    Start with a 8 by 8 board of squares (for instance, a chessboard) and take away the top-left and bottom-right corner squares so that there are 62 squares left. Take some dominoes that are the same size as two squares of the chessboard. Can you cover the 62 squares of the board with 31 such...
  38. C

    Factoring Cubic Equations: What Methods Can Be Used?

    How do I factor this equation x^3-1
  39. P

    Cancelling Squares: Can It Be Done?

    Greetings friends, I have come across an argument on cancelling the squares on either side of an equation. For example if the equation is (a-b)^2=(c-b)^2 my argument is that i can cancel the squares by taking the square root of both sides as to get (a-b)=(c-b) and hence a=c. But others says...
  40. A

    Square Packing Solutions for 24 Integer Squares

    \begin{array}{c} {{A_n}={\sqrt{\sum _{z=1}^{n}{z^2}}} } \\ {{A_1}=1 } \\ {{A_{24}}=70}\end{array}\ Is there a proof that only for n =1 or n=24 that An is an integer quantity?
  41. S

    Uncovering the Mystery of Magic Squares

    Magic Square Hello, Don't know, which forum, so i put it to general... Yesterday i saw something like an magician on an exposition, showing some math to angle for attention. He asked the audience to give him a number between 41 and 100. So he got the 47. He worked out a magic...
  42. D

    Finding Perfect Squares Modulo a Number: 4 (mod 10) Example

    How do I find out if a number is a perfect square modulo a? For example, is 4 (mod 10) a perfect square?
  43. dextercioby

    Calculating Sums of Reciprocal Squares and Fourth Powers

    Compute the following: \sum_{n=1}^{+\infty} \frac{1}{n^{2}} =...?? \sum_{n=1}^{+\infty} \frac{1}{n^{4}} =...?? .LINKS TO WEBPAGES WITH SOLUTIONS ARE NOT ALLOWED! :-p Daniel.
  44. P

    Linear polynomial least squares

    Construct the normal equations for the linear polynomial least squares to fit the data x = [1 0 -1], y=[3;2;-1]. (a) Find the parameters of the linear regression u1, u2 using QR decomposition, and plot the data and the fit curve in a graph (paper and pencil). (b) Calculate the eigenvalues of the...
  45. B

    Finding co-ordinates in squares

    urgent! finding co-ordinates in squares A(7,2) and C(1,4) are vertices of a square ABCD. equation BD is y=3x-9 midpoints AC: (4,3) find the co-ordinates of B and D i just don't understan how to get the co-ordinates, I've tried plotting a graph but to no avail. i found the midpoints...
  46. X

    Calculating Distance Between Two Points: Simplifying Radical Expressions

    First I'd like to say that I'm getting back into college after several years out in the job market. Unfortunately, I need to complete several more upper division math courses before I can complete my CS degree. Before I go back and start taking my classes again, I've been trying to self-study...
  47. T

    Sums of digits of Perfect Squares

    A little while ago I noticed a pattern in the sums of the digits of perfect squares that seems to suggest that: For a natural number N, the digits of N^2 add up to either 1, 4, 7, or 9. ex: 5^2 = 25, 2+5 = 7 In some cases, the summation must be iterated several times: ex: 7^2 = 49...
  48. R

    Solving the Sum of Squares of 3 Consecutive Odd Numbers

    I'm getting confused and can't seem to wrap my head around this problem. Prove that the sum of the squares of any 3 consecutive odd numbers when divided by 12 gives a remainder of 11. I'm not sure how to set this up or proceed I figured that (n^2 + (n +2)^2 + (n+4)^2)/12 = x + 11...
  49. O

    Searching for Perfect Squares using the Pythagorean Theorem

    perfect square? hello everyone! i have come across an on-going research onDETERMINING A PERFECT SQUARE GIVEN A DIFFERENCE . However, I have a feeling that this was not an original one. the researcher used the Pythagorean theorem to arrive at his so called "theorem". would anyone give...
  50. P

    Welcome to the Magical World of Magic Squares

    Greetings, I suppose all of us have at one time or another been fascinated by "magic squares" My question is: has the relationship of numbers in a magic square been found to be useful in the mathematical sciences in any advanced analytical work? Or is is just a mathematical curiousity?
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