Big Square composed of Small Squares ?

[Albert1]

View attachment 580

 

[soroban]

Hello, Albert!

Prove that it is impossible for a square to be composed
of five smaller square as shown.

         a       b
      *-----*---------*
      |     |         |
    a |     |         | b
      |     |   Q     |
      |    P*---*-----*
      |     |   |     |
      *-----*---*R    |
      |     S   |     |
      |         |     | c
    d |         |     |
      |         |     |
      |         |     |
      *---------*-----*
           d       c

The four outer squares have sides $a,b,c,d$ as shown.

The inner square is $PQRS$.We find that: .$\begin{Bmatrix}PQ \:=\:b-c \\ SR \:=\:d-a \end{Bmatrix} \quad \begin{Bmatrix}QR \:=\:c-d \\ PS \:=\:a-b \end{Bmatrix}$Since $PQ = SR\!:\:b-c \:=\:d-a \quad\Rightarrow\quad a+b-c-d \:=\:0\;\;[1]$

Since $PS =QR\!:\:a-b \:=\:c-d \quad\Rightarrow\quad a-b-c+d \:=\:0\;\;[2]$Add [1] and [2]: .$2a-2c\:=\:0 \quad\Rightarrow\quad a \:=\:c$

Subtract [1] and [2]: .$2b-2d \:=\:0 \quad\Rightarrow\quad b \:=\:d$Hence, the large square is divided into four congruent squares.

The inner square has zero area.

 

[Nono713]

The inner square has zero area.

Still a square! (Tongueout)

 

[Albert1]

Hence, the large square is divided into four congruent squares.
The inner square has zero area. (Tongueout)

soroban :well done !

 

[CellCoree]

I would interpret this phrase as a reference to a geometric shape composed of smaller, identical square units. This concept is commonly known as a "tessellation" and has been studied extensively in mathematics and geometry. The arrangement of these smaller squares could take on various patterns and arrangements, creating an interesting and visually appealing structure. It is also worth noting that this concept has practical applications, such as in tiling and flooring designs. Overall, the idea of a "Big Square composed of Small Squares" is a fascinating and well-studied phenomenon in the field of mathematics and geometry.