Question

In: Advanced Math

The nine entries of a 3×3 grid are filled with −1, 0, or 1. Prove that...

The nine entries of a 3×3 grid are filled with −1, 0, or 1. Prove that among the eight resulting sums (three columns, three rows, or two diagonals) there will always be two that add to the same number.

Expert Solution

There are possible sums and they are . These are obtained when a "place" (column,row or diagonal) contains the entries

i)

ii) (or any permutation of it)

iii) (or any permutation of it)

iv)

v) (or any permutation of it)

vi) (or any permutation of it)

vii)

There are "places" to be filled ( columns, rows and diagonals). Thus by pigeonhole principle, it follows that at least two of the places would contain the same sum.

Colby Messinger answered 1 year ago

2. Prove that |[0, 1]| = |(0, 1)|

1. Prove that it is impossible to have a group of nine people at a party...

1. Prove that it is impossible to have a group of nine people at a party such that each one knows exactly five others in the group. 2. Let G be a graph with n vertices, t of which have degree k and the others have degree k+1. Prove that t = (k+1)n - 2e, where e is the number of edges in G. 3. Let G be a k-regular graph, where k is an odd number. Prove that the...

3) A polystyrene container is filled to the brim with water at 0 0C and sealed...

3) A polystyrene container is filled to the brim with water at 0 0C and sealed with a polystyrene lid of the same thickness as the container. The sealed container is immersed in a reservoir of liquid Nitrogen at a temperature of 73 K. The surface area of the container is 5 x 10-2 m2 and the rate of heat flow from the inside to the outside of the container is 66 Watts. (a) Assuming the temperature of inner and...

Prove that the 0/1 KNAPSACK problem is NP-Hard. (One way to prove this is to prove...

Prove that the 0/1 KNAPSACK problem is NP-Hard. (One way to prove this is to prove the decision version of 0/1 KNAPSACK problem is NP-Complete. In this problem, we use PARTITION problem as the source problem.) (a) Give the decision version of the O/1 KNAPSACK problem, and name it as DK. (b) Show that DK is NP-complete (by reducing PARTITION problem to DK). (c) Explain why showing DK, the decision version of the O/1 KNAPSACK problem, is NP-Complete is good...

[Jordan Measure] Could you prove the following ? Prove that the sets Q ∩ [0, 1]...

[Jordan Measure] Could you prove the following ? Prove that the sets Q ∩ [0, 1] and [0, 1] \ Q are not Jordan measurable.

Prove that the function defined to be 1 on the Cantor set and 0 on the...

Prove that the function defined to be 1 on the Cantor set and 0 on the complement of the Cantor set is discontinuous at each point of the Cantor set and continuous at every point of the complement of the Cantor set.

1. Prove by contraction that L = {0^x 1^y 0^x+y | x >= 1 and...

1. Prove by contraction that L = {0^x 1^y 0^x+y | x >= 1 and y >=1} is not Regular. Must use the Pumping Lemma. [ HINT: Describe the language in English first. Use example 3 from the lecture notes ] a) Choose one S from L where S is longer than N (Describe S in terms of N and M) S = **??** b) List all places v can be in S: (i.e. all possible...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

3. In R4 , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0,...

3. In R4 , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0, 0),(0, 0, 1, 1)}, span R4? In other words, can you write down any vector (a, b, c, d) ∈ R4 as a linear combination of vectors in the given set ? Is the above set of vectors linearly independent ? 4. In the vector space P2 of polynomials of degree ≤ 2, find explicitly a polynomial p(x) which is not in the span...

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are...

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are 1, ?4 and 3. express the equation of the surface x^2 ? 2y^2 + z^2 + 6xy ? 2yz = 16. How should i determine the order of the coefficient in the form X^2/A+Y^2/B+Z^2/C=1?