Question

Two squares of maximum size were obtained from one rectangle

Answer 1

Let, L units be the long side or length of the triangle and B units be the short side or breadth of a rectangle.

So, area of the rectangle would be A=LB(unit)2

Now, as far as your question is concerned, there would arise following cases:

Case-1: When area of each square is 1 sq. units.

Since, area of the rectangle is LB(unit)2 .

This means area of this rectangle can be filled completely by LB number of squares such that every square has area equal to 1(unit)2

Case-2: When area of each square is x sq. units

We know that, area of the rectangle is, A=LB(unit)2 .

This can be written as, A=LBx×x(unit)2

This means area of this rectangle can be filled completely by LBx number of squares such that every square has area equal to x(unit)2

Case-3: When a rectangle is filled completely by maximum number of biggest possible squares

The algorithm I am representing below can be easily implemented in MS excel spreadsheet by implementing following simple relations as formulas and we can easily get number of squares in the rth stage of ‘square estimation' of the given rectangle.

You can easily understand following algorithm by drawing rectangle having any length and breadth of your choice.

Let, Lr and Br represents length and breadth of the rectangle at the rth number of set of squares. So, length of each side of the square in rth set be obviously Br .

Let, nr be the quantity of squares in this rth number of set of squares.

Set-1: r=1

In this case, we have available rectangle with length L1=L0 and breadth B1=B0 .

This rectangle will contain n1 number of squares such that they have largest possible area. Also, it can be observed that length of each side of such squares would be equal to breadth of the rectangle (which is containing these squares).

So, length of each side of the square happens to be B1 .

We also find that, n1=⌊L1B1⌋

So, after cutting these n1 number of first set of squares from the given rectangle , we'll be left with another rectangle whose length is equal to B1 and breadth is equal to (L1−n1B1) . This rectangle will be used to get the second set of number of squares.

Set-2: r=2

From above discussion, it follows that, to proceed in this set, we have available rectangle with length L2=B1 and breadth B2=L1−n1B1 .

This rectangle will contain n2 number of squares such that they have largest possible area and measure of each side equal to B2 .

We also find that, n2=⌊L2B2⌋

So, after cutting these n2 number of second set of squares from the remaining rectangle , we'll (may or may not) still be left with another rectangle whose length is equal to L3 and breadth is equal to B3 such that, L3=B2 and B3=L2−n2B2 .

This rectangle will be used to get the third set of number of squares.

Set-r: r=r

From the above discussion, we find following iterative type of relationships at the rth number of set of rectangle:

In this set, we have available rectangle with length Lr=Br−1 and breadth Br=Lr−1−nr−1Br−1 .

This rectangle will contain nr number of squares such that they have largest possible area and measure of each side equal to Br .

We also find that, nr=⌊LrBr⌋

This rectangle will be used to get the next set of number of squares.

Proceeding this way, we can get finite or infinite number of squares that can be fitted inside the given rectangle.

So, in general, area of the rectangle can be expressed as sum of infinite number of square as follows:

L0B0=∑∞r=1nr(Br)2