Why do 9+4 work? Why does it give you 6?

**loutja35** · September 8th 2010, 01:55 PM

The question is:

How can you bring up from a river, exactly 6 quarts of water, when you have only two containers, a four-quart pail and a nine-quart pail, to measure with. I understand how to get the answer to this but my question is why do 9 quart and 4 quart pails work? Why does it give you 6 quarts when you solve this question?

**pickslides** · September 8th 2010, 02:29 PM

Using the right combination $2\times 9 - 3\times 4$ works. But what are you really asking?

Do you want to know some combinations that don't work? Or you you looking for a generalisation?

**loutja35** · September 8th 2010, 04:45 PM

Thank you for your time. Yes I am looking for a generalization if you have any thoughts.

**skeeter** · September 8th 2010, 06:13 PM

this is like one of those "puzzle" problems. I assume all you have is one each 9qt and 4qt container.

start with 9qt empty

fill 4qt ... dump into 9qt

fill 4qt again ... dump into 9 qt

fill 4qt again ... top off 9qt. this leaves 3 qts in the 4 qt container.

empty the 9qt ... pour the 3 qts into the 9qt from the 4qt container.

fill 4qt ... pour into 9qt. now have 3+4 = 7 qts in the 9qt container.

fill 4qt ... top off 9qt container. leaves 2 qts in the 4qt container.

empty 9qt ... pour the 2qts into the 9qt container.

fill the 4qt ... pour all 4 qts into the 9qt. 2 + 4 = 6 qts in the 9 qt container.

**undefined** · September 8th 2010, 07:11 PM

Originally Posted by loutja35

The question is:

How can you bring up from a river, exactly 6 quarts of water, when you have only two containers, a four-quart pail and a nine-quart pail, to measure with. I understand how to get the answer to this but my question is why do 9 quart and 4 quart pails work? Why does it give you 6 quarts when you solve this question?

One generalisation going along w/ skeeter's reply is: If you have two pails, one that holds x quarts where x is a positive integer, and one that holds kx + 1 quarts for k a positive integer, you will be able to measure n quarts for any n in {0, ... , (k+1)x + 1}.

**loutja35** · September 9th 2010, 01:19 AM

This is wonderful. Thank you very much. This is what I was trying to understand.

**melese** · September 9th 2010, 01:56 AM

Here's a generalization: Suppose you have two pails, one that can hold exactly $a$ quarts of water and another that can hold exactly $b$ quarts of water.
Furthermore, $a$ and $b$ are relatively prime and $a<b$ . In this case, you can measure out any integer number of quarts $0\leq n\leq b$ of water, in the $b$ -quarts pail.

The way to do this parctically is described by the procedure skeeter gave.

**undefined** · September 9th 2010, 04:54 AM

Originally Posted by melese

Here's a generalization: Suppose you have two pails, one that can hold exactly $a$ quarts of water and another that can hold exactly $b$ quarts of water.
Furthermore, $a$ and $b$ are relatively prime and $a<b$ . In this case, you can measure out any integer number of quarts $0\leq n\leq b$ of water, in the $b$ -quarts pail.

The way to do this parctically is described by the procedure skeeter gave.

I wanted to say this, but made a silly error mentally checking {9,5} and thought it somehow failed! Adding and subtracting single digit numbers can be challenging... especially late at night.

**Soroban** · September 9th 2010, 12:05 PM

Hello, loutja35!

I too am puzzled by your question.
But I'll give it a try . . .

How can you measure 6 quarts of water, when you have only two containers:
a 4-quart pail and a 9-quart pail to measure with?

My question is: why do 9 quart and 4 quart pails work?
Why does it give you 6 quarts when you solve this question?

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 $A$ = 9-quart pail.
Let $B$ = 4-quart pail.

Fill $\,A.$

Code:

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

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

Code:

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

Empty $\,A.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Fill $\,B.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Fill $\,B.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Empty $\,A.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Fill $\,B.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Fill $\,B.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Empty $\,A.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Fill $\,B.$

Code:

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

Pour $\,B$ into $A.$

Code:

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

Fill $\,B.$

Code:

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

Pour $\,B$ into $A.$

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?"

**loutja35** · September 12th 2010, 07:05 PM

Could you actually further explain your reasoning and how you decided on kx+1. What exactly does this stand for? Also how did you come up with (o,....,(k+1)x+1). How does this specifically work? Could you show me an exampe of this. As always, thank you very much!

**undefined** · September 12th 2010, 07:24 PM

Originally Posted by loutja35

Could you actually further explain your reasoning and how you decided on kx+1. What exactly does this stand for? Also how did you come up with (o,....,(k+1)x+1). How does this specifically work? Could you show me an exampe of this. As always, thank you very much!

For {9,4} note that at the beginning you fill the 4 quart pail to the top, so there are 4 quarts in it.

After topping the 9 quart pail off, you are left with 3 quarts in the 4 quart pail.

After topping the 9 quart pail off again, you are left with 2 quarts in the 4 quart pail.

After topping the 9 quart pail off again, you are left with 1 quart in the 4 quart pail.

This allows you to get any number of quarts in the 9 quart pail up to 9 quarts. (Think about why.)

Note that if the large pail has capacity of any number in the set {5,9,13,17,21,...} the effect is the same; we can get any number of quarts up to the capacity of the pail. (Think about why.)

Furthermore the same thing happens whenever the small pail is x and the larger pail is a (positive) multiple of x plus 1. (Think about why.)

Furthermore there is no need for the number of quarts left in the small pail to be the sequence x, x-1, ... , 1. As long as every number in {1, 2, ... , x} can be obtained in the small pail, any number up to capacity of large pail can be gotten in large pail. This is true if and only if gcd(x, y) = 1 where y is the capacity of larger pail.

**undefined** · September 12th 2010, 08:35 PM

A more advanced/thorough treatment can be found in this thread

http://www.mathhelpforum.com/math-he...em-155981.html

Thread: Why do 9+4 work? Why does it give you 6?

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