Tricky Math Problems: Solving Square and Rectangle Challenges

[katek8k8]

Homework Statement

1. Given a 5 by 5 square with the numbers 1,2,3,4,...,25 in sequence in the successive rows. Pick out five numbers so that no two of them are in the same row or same column. What is their sum? Prove that this sum is constant for any five numbers chosen this way.



2. The points A(5,6) and C(3,0) are opposite vertices of rectangle ABCD. The other two vertices B and D, lie on the line y=3, with B having a positive abscissa. Find the coordinates of vertices B and D.

Homework Equations

First problem uses addition.

Second problem uses an equation having to do with abscissas?

 

[Mark44]

Homework Statement

1. Given a 5 by 5 square with the numbers 1,2,3,4,...,25 in sequence in the successive rows. Pick out five numbers so that no two of them are in the same row or same column. What is their sum? Prove that this sum is constant for any five numbers chosen this way.



2. The points A(5,6) and C(3,0) are opposite vertices of rectangle ABCD. The other two vertices B and D, lie on the line y=3, with B having a positive abscissa. Find the coordinates of vertices B and D.

Homework Equations

First problem uses addition.

Second problem uses an equation having to do with abscissas?

For 2, abscissa just means x coordinate here. For the unknown points B and D, the y coordinate is 3 for each. From the given information, the sides of the rectangle aren't parallel to either axis.

In a rectangle, the diagonals are equal in length, and the adjacent sides have to be perpendicular. Use these facts to get equations that involve the unknown x coordinates.

 

[Mark44]

For 1, the numbers 1, 2, 3, ..., 24, 25 are laid out in order, in the rows of the 5 x 5 matrix. Each number can be associated with its row (rows 0 through 4) and column (columns 1 through 5) with this formula: num = 5 * row number + col number. For example, 17 is in row 3, column 2, and 17 = 5 * 3 + 2.

Pick five numbers from the array and add them.
Total =
5 * row(i₁) + col(j₁) +
5 * row(i₂) + col(j₂) +
5 * row(i₃) + col(j₃) +
5 * row(i₄) + col(j₄) +
5 * row(i₅) + col(j₅)

If you pick the numbers according to the instructions in this problem, what do you get for the sum?