There are six bowls in a circle There is one frog in each bo

There are six bowls in a circle. There is one frog in each bowl. Each frog is able to jump into an adjacent bowl but no further. Every minute, two frogs simultaneously jump into new bowls.(This may result in a bowl containing more than one frog.) Is it possible that at some point all of the frogs will be in the same bowl?

Solution

No it is not possible as if we fix a frog in some bowl say 6th then after 1 minute say the frogs from the 1st bowl and the 5th bowl jump in the 6th bowl.

In that case after one minute there will be 3 frogs in the 6th bowl and no frog in the 1st and 5th bowl while other bowls will have one frog each .

After the second minute let the frog from 2nd bowl jump in 1st bowl and the frog from the 4th bowl jump in the 5th bowl as only two frogs jump after every minute.

Then after two minutes there are 3 frogs in 6th bowl and 1 frog in 1st , 5th and 3rd bowl and no frog in the 2nd and 4th bowl.

After 3rd minute let the frogs jump from the 1st bowl and from the 5th bowl in the 6th bowl

So that there are 5 frogs in 6th bowl and no frog in the 1st,2nd, 4th and 5th bowl while 1 frog in 3rd bowl.

Now since after evry minute two frogs jump so frog from the 3rd bowl may come to 6th bowl but at the same time other frogs will have shifted from the 6th bowl.

Hence it is never possible under the given conditions that one bowl will contain all the six frogs