Question 159
A polymino is a sequence of squares $1 \times 1$ juxtaposed by sharing sides together with their neighbors and forming a single piece. The two-square polymers are known as dominoes and the four-square polymers are known as tetraminos, the pieces of the famous Tetris game. The following figure shows a square $3 \times 3$ with numbers from 1 to 9 written on each of its squares $1 \times 1$.
| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
Knowing that $1+2+\ldots+9=45$ we can try to divide the square into 3 polyminos with the same sum, each with a sum $\frac{45}{3}=15$. The following figure shows a way to do this.
| 1 | 2 | 3 |
| :---: | :---: | :---: |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
a. Show that it is not possible to divide the square $3 \times 3$ into an amount greater than three of the same sum-polyminoes.
b. Consider the square $4 \times 4$ with the numbers from 1 to 16, written in ascending order as shown in the figure below
| 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: |
| 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 |
| 13 | 14 | 15 | 16 |
Shows how to divide this square into two polymers so that the sum of the numbers in each of them is the same.
c. Consider the square $5 \times 5$ with the numbers from 1 to 25 written in ascending order following the pattern in the previous figure. Show that it is not possible to divide this square into two or more polymers with the same sum of the numbers in each of them.
| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
Knowing that $1+2+\ldots+9=45$ we can try to divide the square into 3 polyminos with the same sum, each with a sum $\frac{45}{3}=15$. The following figure shows a way to do this.
| 1 | 2 | 3 |
| :---: | :---: | :---: |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
a. Show that it is not possible to divide the square $3 \times 3$ into an amount greater than three of the same sum-polyminoes.
b. Consider the square $4 \times 4$ with the numbers from 1 to 16, written in ascending order as shown in the figure below
| 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: |
| 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 |
| 13 | 14 | 15 | 16 |
Shows how to divide this square into two polymers so that the sum of the numbers in each of them is the same.
c. Consider the square $5 \times 5$ with the numbers from 1 to 25 written in ascending order following the pattern in the previous figure. Show that it is not possible to divide this square into two or more polymers with the same sum of the numbers in each of them.