Question

A country road is 27 miles long and goes all the way around a lake, connecting the six cottages that

are next to the lake. Two of the cottages are 1 mile apart (along the road). Two cottages are 2 miles
apart, two are 3 miles apart, two are 4 miles apart..... two are 25 miles apart, and two are 26 miles
apart. How are the cottages distributed along the road? Find a second way to distribute
them. Explain your thinking.

Answer 1

Explanation:

The total length of the road is 27 miles, and since each cottage is separated from its neighbor by a distance of 1, 2, 3, 4, ..., 25, or 26 miles, the total length of the road occupied by these distances must sum to 27 miles. This is because the road connects the six cottages.

One possible distribution is:

Cottage 1 and 2 are 1 mile apart.

Cottage 3 and 4 are 2 miles apart.

Cottage 5 and 6 are 3 miles apart.

Cottages 1 and 3 are 4 miles apart.

Cottages 2 and 4 are 5 miles apart.

Cottages 3 and 5 are 6 miles apart.

Cottages 4 and 6 are 7 miles apart.

...

Cottages 1 and 6 are 25 miles apart.

Cottages 2 and 5 are 26 miles apart.

A second possible distribution can be obtained by reversing the order of the distances between the cottages. That is:

Cottages 2 and 5 are 26 miles apart.

Cottages 1 and 6 are 25 miles apart.

Cottages 4 and 6 are 7 miles apart.

Cottages 3 and 5 are 6 miles apart.

Cottages 2 and 4 are 5 miles apart.

Cottages 1 and 3 are 4 miles apart.

Cottage 5 and 6 are 3 miles apart.

Cottage 3 and 4 are 2 miles apart.

Cottage 1 and 2 are 1 mile apart.

There may be other possible distributions, but these two demonstrate that there are multiple ways to arrange the distances between the cottages while still satisfying the constraint that the distances must sum to 27 miles.