Did you solve it? Are you smarter than a forester?

One, two, tree Photograph: jin_tang/Getty Images/iStockphoto

Earlier today in my puzzle blog I asked the following problems about planting trees on an island:

(For the purposes of this puzzle the island is empty apart from the trees, and a tree is hidden only when it lies directly behind another tree from the perspective of the observer).

Problem 1

The island has 5 trees positioned at the vertices of a regular pentagon, as below. Is it possible for you and two friends to stand on the island, so that the three of you can each see a different number of trees?

Solution

Yes it is possible. Here are three positions, from which you can see 3, 4 and 5 trees.

Problem 2

What is the highest possible number of people that can stand on the island, with each person seeing a different number of trees, when the island has 6 trees ? Design such an arrangement of trees.

Solution

The highest number of people we could possibly hope for would be 6, who are able to see 1, 2, 3, 4, 5 and 6 trees respectively. However, we can show this is impossible by considering that the only way it would be possible for a person to see a single tree would be if the trees are all on the same line, as below. But if they are there is no position from which you can see exactly 3 trees.

So, the answer is at most 5. And if you sketched a few arrangements you may have found one with five, such as this one here. (There are others too). The answer is 5.

The trees are arranged on two perpendicular lines. Each numbered black dot shows a position from where you can see that number of trees.

Thanks again to Daniel Griller for this puzzle. Check out his book Elastic Numbers.

Can You Solve My Problems

I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

My puzzle book Can You Solve My Problems? is just out in paperback.