Counting and Abstract Problem Solving

Question

Suppose that you have a bucket holds fiv-sev c, and one holds tw-one c. How could you use them to measure out thre c of water?

Answer 1

Basically, you want to think about how you can make $3$ as a linear combination of $57$ and $21$. However, you can't have negative numbers anywhere in this process.

Essentially, what you want to do is find minimal integers $x$ and $y$ such that $21x-57y=3$ or such that $57y-21x=3$. After you do this, you simply want to translate this addition and subtraction into physical processes such as dumping out the water, filling buckets, etc.

1. Fill the $57$-cup bucket.
2. Pour $21$ cups into the smaller $21$-cup bucket.
3. Empty the smaller bucket.
4. Repeat 2 & 3.
5. Pour the remaining $15$ cups into the smaller bucket.
6. Fill the $57$-cup bucket.
7. Pour $6$ cups into the smaller $21$-cup bucket so that it is full.
8. Empty the small bucket. There are $51$ cups in the large bucket now.
9. Pour $21$ cups into the smaller $21$-cup bucket.
10. Empty the smaller bucket.
11. Repeat 9 & 10.
12. Pour the remaining $9$ cups into the smaller bucket.
13. Fill the $57$-cup bucket.
14. Pour $12$ cups into the smaller $21$-cup bucket so that it is full.
15. Empty the small bucket. There are $45$ cups in the large bucket now.
16. Pour $21$ cups into the smaller $21$-cup bucket.
17. Empty the smaller bucket.
18. Repeat 9 & 10.

You should be left with $3$ cups in the large bucket.