# Big Square composed of Small Squares ?

#### soroban

##### Well-known member
Hello, Albert!

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

MHB Math Helper

#### Albert

##### Well-known member
Hence, the large square is divided into four congruent squares.
The inner square has zero area.
soroban :well done !