After a huge storm hits the Hex Archipelago, you come across five people floating in the water. They are cold and wet, and they tell you they were part of the crew of the famous pirate Greenbeard. They explain that Greenbeard left them stranded on an island after they tried to overthrow him. Each of them was tied up in a different spot on the island until the storm swept them away. To thank you for saving them, they share a secret: the island they were on is where Greenbeard hid his treasure. But when they try to describe the island, things get confusing. They all agree the island was flat and empty except for some trees, but each pirate says they saw a different number of trees, from two to six. The pirate who saw two trees is sure the treasure was buried right at his feet.
You decide to fly over the area in your hot air balloon to check it out. You see hundreds of small islands, each with exactly six trees. With another storm on the way, you need to quickly figure out which island has the treasure. What does the island with Greenbeard’s treasure look like from above? And where exactly is the treasure buried on that island?
At first, it might seem like the pirates are just confused, but that’s not true. Each pirate was stuck in a different spot on the island, and none of them could see the same number of trees. This means something was blocking their view, and that something was the other trees. A pirate would see fewer trees if two or more trees lined up from where they were standing. So, we need to find the island where five different pirates, each in a different spot, would see a different number of trees.
Most islands have a spot where you can see all six trees, and many have a spot where you can see five trees if two trees are lined up. The tricky part is finding spots where fewer trees are visible because that requires more trees to line up perfectly.
To see just two trees, imagine if all the trees were in a straight line. You could stand at one end and see one tree, stand in the middle and see two, or stand anywhere else and see all six. But you can’t see only three, four, or five trees this way, so a straight line won’t work.
What if there are two lines of trees? As long as these lines aren’t parallel and they cross each other on land, there will be a spot where the lines meet, allowing you to see exactly two trees. If the trees are grouped in twos and fours, or threes and threes, there are many ways you could see three, four, five, and six trees.
Luckily, there’s only one island in the archipelago with two non-parallel lines of trees. The treasure is buried where these lines intersect. You land on this island and dig up a chest filled with a large pile of tree seeds, ready for planting. Was this treasure worth all the effort? That depends on how you look at it!
Using clay or playdough, create a model of the island described in the article. Arrange small sticks or toothpicks to represent the trees. Try to position them in two non-parallel lines, as mentioned in the solution. This will help you visualize how the pirates saw different numbers of trees from different spots.
Draw a top-down view of the island with six trees. Mark different spots where the pirates might have stood. From each spot, draw lines of sight to the trees they could see. Label each spot with the number of trees visible from there. This will help you understand how the pirates’ perspectives differed.
Set up a classroom or outdoor area with objects representing trees. Blindfold a volunteer and guide them to different spots. Have them describe how many “trees” they can “see” based on touch or sound. Discuss how this relates to the pirates’ experience and the concept of perspective.
Rewrite the story from the perspective of one of the pirates. Describe their experience on the island and how they determined the number of trees they could see. Share your story with the class to explore different viewpoints and interpretations.
Using protractors and rulers, draw two non-parallel lines on graph paper to represent the tree lines. Calculate the angles formed where the lines intersect. Discuss how angles and geometry play a role in determining visibility and perspective, similar to the pirates’ situation.
After a massive storm tears through the Hex Archipelago, you find five survivors in the water. Shivering, they explain that they were part of the crew of the great pirate Greenbeard, who marooned them after a failed mutiny. Each was bound in a different spot on a small island until the storm washed them out to sea. In gratitude for being rescued, they reveal a secret: the island they were on is where Greenbeard buried his treasure. However, when they try to describe the island, something seems off. All agree it was flat and barren with no prominent features except for some trees, yet each pirate claims to have seen a different number of trees, ranging from two to six. The pirate who saw two trees insists the treasure was buried right at his feet.
When you fly your hot air balloon over the area to investigate, you see hundreds of small islands, each with exactly six trees. With another storm approaching, you’ll need to hurry and narrow your search. What does the island with Greenbeard’s treasure look like from the sky? And where will the treasure be on that island?
It might seem like the pirates are confused from dehydration, but that’s not the case. Each was confined to a separate point on the island, and no two of them could see the same number of trees. This means that for all but one pirate, something was blocking their view, which could only have been other trees. A pirate would see fewer trees when two or more were aligned from their vantage point. Thus, we need to find the island where five different pirates standing in different spots would each see a different number of trees.
Most islands have a position from which you can see six trees, and many have a position where five trees can be seen by standing in line with two of them. The hardest locations to find are those with fewer visible trees, as they require more trees to align with the viewer’s position.
To see just two trees, one possibility is if all the trees were lined up in a single file. In that case, you could stand at the end and see one, in the middle and see two, or anywhere else and see all six. However, there’s no position from which you can see only three, four, or five, so a straight line of trees won’t work.
What about two lines of trees? As long as the lines aren’t parallel and intersect over land, there will always be a position where the two lines converge, allowing you to see exactly two trees. If they’re grouped two and four, or three and three, there are many arrangements in which you could also see three, four, five, and six trees.
Fortunately, there’s only one island in the archipelago with two non-parallel lines of trees, and the treasure will be buried at the intersection of those lines. You land on this island and dig up a chest containing a large pile of tree seeds, ready for planting. Was this treasure really worth all that trouble? That’s a matter of perspective.
Island – A distinct area or shape within a plane that is surrounded by other shapes or empty space. – In geometry class, we learned how to calculate the area of an island formed by intersecting circles on a graph.
Trees – Branching diagrams used to represent hierarchical relationships, often used in probability and statistics. – We used trees to map out all possible outcomes of flipping a coin three times.
Treasure – A valuable solution or result obtained after solving a complex mathematical problem. – After working through the challenging geometry problem, finding the correct answer felt like discovering a hidden treasure.
Pirates – Imaginary characters used in math problems to add context, often involving sharing or dividing resources. – The pirates had to divide their treasure equally, which helped us understand the concept of division in math.
View – A perspective or angle from which a geometric figure is observed or analyzed. – From the top view, the 3D shape appeared as a simple rectangle.
Lines – Straight one-dimensional figures that extend infinitely in both directions, used to define shapes and angles. – In geometry, we learned how parallel lines never intersect, no matter how far they are extended.
Buried – Hidden or embedded within a complex problem or figure, often requiring deeper analysis to uncover. – The solution to the equation was buried within a series of complex steps.
Spot – A specific point or location on a geometric figure or graph. – We had to find the exact spot where the two lines intersected on the coordinate plane.
Number – A mathematical object used to count, measure, and label, fundamental to all mathematical operations. – The number of sides on a polygon helps determine its classification in geometry.
Geometry – The branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. – In geometry, we explore different shapes and learn how to calculate their areas and volumes.
