Problem No. 20 A cup of water

"Please give me a cup of water!"

You have a 7 cup bottle and a 30 cup bottle. How can you measure one cup of water?

Parent tip

You can make your own 7 cup and 30 cup bottles. Take two bottles, measure the wanted amount of water and mark the bottles, very carefully, at their maximal levels, the first one as the 7 cup bottle, and the other one as the 30 cup bottle. Then you can all have fun, you and your children, with pouring the water in and out of the bottles.

Solution

If we denote by [x,y] that in the 7 cup bottle there are x cups of water, and in the 30 cup bottle there are y cups of water, then at the beginning we have the [0,0] situation. We can fill any bottle to its top at each time and we can pour from one bottle to other either the whole amount of water (if there is a space for it), or the amount of water that fills the other bottle to its top. If at some instance one of the values drop to 0, it means that the water from that bottle is poured away. One of the possible solutions for measuring a cup of water is:

[7,0]→[0,7]→[7,7]→[0,14]→[7,14]→[7,0]→[0,7]→[7,7]→[0,14]→[7,14]→
→[0,21]→[7,21]→[0,28]→[7,28]→[5,30]→→[0,21]→[7,21]→[0,28]→[7,28]→
→[5,30]→[5,0]→[0,5]→[7,5]→[0,12]→[7,12]→→[5,0]→[0,5]→[7,5]→[0,12]→
→[7,12]→[0,19]→[7,19]→[0,26]→[7,26]→[3,30]→→[0,19]→[7,19]→[0,26]→
→[7,26]→[3,30]→[3,0]→[0,3]→[7,3]→[0,10]→[7,10]→→[3,0]→[0,3]→[7,3]→
→[0,10]→[7,10]→[0,17]→[7,17]→[0,24]→[7,24]→[1,30]→[0,17]→[7,17]→
→[0,24]→[7,24]→[1,30]

And in the 7 cup bottle there is a cup of water left.