Answer: because a 4-quart pail and a 9-quart pail can be used

. . . . . . to produce *any* integer quantity from 1 quart to 9 quarts.

Let $\displaystyle A$ = 9-quart pail.

Let $\displaystyle B$ = 4-quart pail.

Fill $\displaystyle \,A.$

Code:

* *
|:::|
|:::|
|:::| * *
|:9:| | | We have 9 quarts.
|:::| | |
|:::| | |
|:::| | |
*---* *---*
A B

Pour $\displaystyle \,A$ into $\displaystyle \,B.$

Code:

* *
| |
| |
|:::| * *
|:::| |:::| We have 5 quarts.
|:5:| |:::|
|:::| |:4:|
|:::| |:::|
*---* *---*
A B

Empty $\displaystyle \,A.$

Code:

* *
| |
| |
| | * *
| | |:::|
| | |:::| We have 4 quarts.
| | |:4:|
| | |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
| |
| |
| | * *
|:::| | |
|:::| | |
|:4:| | |
|:::| | |
*---* *---*
A B

Fill $\displaystyle \,B.$

Code:

* *
| |
| |
| | * *
|:::| |:::|
|:::| |:::|
|:4:| |:4:|
|:::| |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
| |
|:::|
|:::|
|:::| * *
|:8:| | | We have 8 quarts.
|:::| | |
|:::| | |
*---* *---*
A B

Fill $\displaystyle \,B.$

Code:

* *
| |
|:::|
|:::| * *
|:8:| |:::|
|:::| |:::|
|:::| |:4:|
|:::| |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
|:::|
|:::|
|:::| * *
|:9:| | |
|:::| |:::| We have 3 quarts.
|:::| |:3:|
|:::| |:::|
*---* *---*
A B

Empty $\displaystyle \,A.$

Code:

* *
| |
| |
| | * *
| | | |
| | |:::|
| | |:3:|
| | |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
| |
| |
| | * *
| | | |
|:::| | |
|:3:| | |
|:::| | |
*---* *---*
A B

Fill $\displaystyle \,B.$

Code:

* *
| |
| |
| | * *
| | |:::|
|:::| |:::|
|:3:| |:4:|
|:::| |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
| |
| |
|:::| * *
|:::| | |
|:7:| | | We have 7 quarts.
|:::| | |
|:::| | |
*---* *---*
A B

Fill $\displaystyle \,B.$

Code:

* *
| |
| |
|:::| * *
|:::| |:::|
|:7:| |:::|
|:::| |:4:|
|:::| |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
|:::|
|:::|
|:::| * *
|:9:| | |
|:::| | | We have 2 quarts.
|:::| |:::|
|:::| |:2:|
*---* *---*
A B

Empty $\displaystyle \,A.$

Code:

* *
| |
| |
| | * *
| | | |
| | | |
| | |:::|
| | |:2:|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
| |
| |
| | * *
| | | |
| | | |
|:::| | |
|:2:| | |
*---* *---*
A B

Fill $\displaystyle \,B.$

Code:

* *
| |
| |
| | * *
| | |:::|
| | |:::|
|:::| |:4:|
|:2:| |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
| |
| |
|:::| * *
|:::| | |
|:6:| | | We have 6 quarts.
|:::| | |
|:::| | |
*---* *---*
A B

Fill $\displaystyle \,B.$

Code:

* *
| |
| |
|:::| * *
|:::| |:::|
|:6:| |:::|
|:::| |:4:|
|:::| |:::|
*---* *---*
A B

Pour $\displaystyle \,B$ into $\displaystyle A.$

Code:

* *
|:::|
|:::|
|:::| * *
|:9:| | | We have 1 quart.
|:::| | |
|:::| | |
|:::| |:1:|
*---* *---*
A B

We have devised a procedure to get 1, 2, 3, 4, 5, 6, 7, and 8 quarts.

. . The problem is *completely solved.*

Why are you asking "Why?"