6,7 |

On six fir trees, six goldfinches are sitting, one on each tree. The trees grow in a row with intervals of 10 meters. If one goldfinch flies from one tree to another, then another goldfinch must necessarily fly the same distance but in the opposite direction.

a) Can all the goldfinches gather on one tree?

b) And if there are seven goldfinches and seven trees?
a) First method. Let's number the trees from 1 to 6 in order. Suppose each chick gets a number equal to the number of the tree it is sitting on (at the moment). Then the sum of the numbers of the chicks is an invariant. Initially, it is equal to \(1+2+\ldots+6=21\). Since 21 is not divisible by 6, the chicks will not be able to gather on one tree.

Second method. Replace the odd-numbered trees with oaks and note that there will always be an odd number of chicks on the oaks.

b) It is not difficult to gather all the chicks on the fourth tree: the chicks from the first and seventh (second and sixth, third and fifth) can fly there simultaneously.

## Answer

a) They cannot; b) they can.