Using the right combination 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?
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?
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.
Here's a generalization: Suppose you have two pails, one that can hold exactly quarts of water and another that can hold exactly quarts of water.
Furthermore, and are relatively prime and . In this case, you can measure out any integer number of quarts of water, in the -quarts pail.
The way to do this parctically is described by the procedure skeeter gave.
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 = 9-quart pail.
Let = 4-quart pail.
Fill
Code:* * |:::| |:::| |:::| * * |:9:| | | We have 9 quarts. |:::| | | |:::| | | |:::| | | *---* *---* A B
Pour into
Code:* * | | | | |:::| * * |:::| |:::| We have 5 quarts. |:5:| |:::| |:::| |:4:| |:::| |:::| *---* *---* A B
Empty
Code:* * | | | | | | * * | | |:::| | | |:::| We have 4 quarts. | | |:4:| | | |:::| *---* *---* A B
Pour into
Code:* * | | | | | | * * |:::| | | |:::| | | |:4:| | | |:::| | | *---* *---* A B
Fill
Code:* * | | | | | | * * |:::| |:::| |:::| |:::| |:4:| |:4:| |:::| |:::| *---* *---* A B
Pour into
Code:* * | | |:::| |:::| |:::| * * |:8:| | | We have 8 quarts. |:::| | | |:::| | | *---* *---* A B
Fill
Code:* * | | |:::| |:::| * * |:8:| |:::| |:::| |:::| |:::| |:4:| |:::| |:::| *---* *---* A B
Pour into
Code:* * |:::| |:::| |:::| * * |:9:| | | |:::| |:::| We have 3 quarts. |:::| |:3:| |:::| |:::| *---* *---* A B
Empty
Code:* * | | | | | | * * | | | | | | |:::| | | |:3:| | | |:::| *---* *---* A B
Pour into
Code:* * | | | | | | * * | | | | |:::| | | |:3:| | | |:::| | | *---* *---* A B
Fill
Code:* * | | | | | | * * | | |:::| |:::| |:::| |:3:| |:4:| |:::| |:::| *---* *---* A B
Pour into
Code:* * | | | | |:::| * * |:::| | | |:7:| | | We have 7 quarts. |:::| | | |:::| | | *---* *---* A B
Fill
Code:* * | | | | |:::| * * |:::| |:::| |:7:| |:::| |:::| |:4:| |:::| |:::| *---* *---* A B
Pour into
Code:* * |:::| |:::| |:::| * * |:9:| | | |:::| | | We have 2 quarts. |:::| |:::| |:::| |:2:| *---* *---* A B
Empty
Code:* * | | | | | | * * | | | | | | | | | | |:::| | | |:2:| *---* *---* A B
Pour into
Code:* * | | | | | | * * | | | | | | | | |:::| | | |:2:| | | *---* *---* A B
Fill
Code:* * | | | | | | * * | | |:::| | | |:::| |:::| |:4:| |:2:| |:::| *---* *---* A B
Pour into
Code:* * | | | | |:::| * * |:::| | | |:6:| | | We have 6 quarts. |:::| | | |:::| | | *---* *---* A B
Fill
Code:* * | | | | |:::| * * |:::| |:::| |:6:| |:::| |:::| |:4:| |:::| |:::| *---* *---* A B
Pour into
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?"
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.
A more advanced/thorough treatment can be found in this thread
http://www.mathhelpforum.com/math-he...em-155981.html