3D Treasure Hunt
2 out of 3
Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?
Problem
This problem follows on from Treasure Hunt, which you may wish to explore first.
Imagine you are looking for treasure in a strange three-dimensional treasure island, where treasure can be hidden at any point on a grid defined by three coordinates.
When you enter a guess in the interactivity below, it tells you the shortest distance you would have to travel along grid lines to reach the treasure.
For example, if the treasure is at (1, 2, 3) and you enter (3, 3, 3), the interactivity will tell you that the treasure is a distance of 3 steps away (2 steps in the x direction, 1 in the y direction and 0 in the z direction).
Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of moves?
Getting Started
If you have not met this type of problem before - solving it in 2D first is a very good idea.
It is worth looking at some very simple cases first and spending some time understanding the distances in 3D.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem offers students an opportunity to explore coordinates in three dimensions, within the engaging context of trying to find some hidden treasure. Students can also test out their strategic thinking by considering efficient ways to find the treasure in the fewest number of guesses.
Possible approach
Introduce the problem by showing the interactivity to the class. You could do this by inputting a guess and then asking students to suggest examples of points which are the required distance from the guess. Explain that the idea is to try to narrow down the possibilities to find the treasure.
Give students some time to work on the problem using the interactivity. If computers or tablets are unavailable, students could instead work in pairs, taking it in turns to choose where the treasure is and work out the distances. Encourage them to keep score of the number of guesses they needed to find the treasure.
Bring the class together and invite them to share useful ideas and efficient strategies. Encourage the students to consider all the points that satisfy each condition, and to look at the surface made in three dimensions by these points.
Key questions
Which points satisfy the conditions given so far?
How can you narrow down the possibilities?
Is it better to guess a point in the middle of the region or at an edge?
Possible support
The challenges in two dimensions in the problem Treasure Hunt provide a useful starting point, and the insights gained can then be extended into three dimensions.
Possible extension
Invite students to represent algebraically the families of points that satisfy the conditions at each step, and to use their geometric interpretation of the algebraic representations to develop an optimal strategy.
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