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.