Treasure hunt
Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?
Problem
Can you find the hidden treasure?
The treasure has been hidden somewhere on this beach, where the grid lines intersect (cross).
Input coordinates to help find the treasure with the fewest guesses. The interactivity gives you the shortest distance you'd have to travel (along the grid lines) to reach the treasure.
Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses? How do you know that your strategy will always work?
Getting Started
Try to decide which places are good for first guesses.
Once you've got feedback on your first guess identify all the possible places where the treasure could be. What pattern does this make?
Have another guess. Use the feedback to identify all the possible places where the treasure could be.
Now use the feedback from both guesses to find some overlapping points.
Continue in the same way...
Student Solutions
Ci Hui Minh Ngoc Ong from Kelvin Grove State College (Brisbane) in Australia, Uday from England and Asiya all sent in their strategies for level 1. This is Uday’s strategy:
You must start in the middle (6,6) as the middle is equally as close to all four of the corners. Then if the treasure is more than (I used this rule, you don't have to use these exact same numbers) 4 steps away, then you go for (any) one of the corners and you input the coordinates right in between the center and the corner you chose. If it's further away than when you did (6,6) go for the midpoint of the center and the opposite corner. Keep doing this until you get close (less than 6 steps away) then you find every possible coordinate it could be but don't enter it yet, click on them and then count if they are the number away from the points previously plotted as it say on the points. When you find one which is correct and you've checked it with all of the points previously plotted, enter it.
Asiya and Ci Hui Minh Ngoc Ong had the same strategy. This is Asiya’s description with Ci Hui Minh Ngoc Ong’s diagrams:
Firstly, I’d choose the spot most close to the centre, this was because I’d have the most space to plot the points around. after making my first guess I plotted all the possible points and realised it made a symmetrical diamond shape, the length and height being the closest possible steps given, I then chose one of the point in the diamond shape and repeat the process, which gives me only two possible overlapping points. I’d guess either of these points completing it usually under 4 guesses.
Ci Hui Minh Ngoc Ong also sent in a strategy for level 2 (click on the image to see a larger version):
This is Aisha’s strategy for level 3:
For level 3 I did the same, however instead of the middle of the graph I’d do it in the middle of the pink space. Creating the diamond shape around id guess two other spaces usually farther from the centre and plot around it which would leave me one point
Elijah and his mum worked out a strategy for level 4:
First of all we had to work out how to think about this problem. My mum worked it out. She tried drawing 3d pictures on squared paper, and then on isometric paper, but they were both too confusing. Then she remembered games of 3d chess and 3d noughts and crosses and so she drew ten 9x9 grids, and labelled them z=0 up to z=9.
First guess: (0, 0, 0) I marked all the possible points that were the right distance away - these came out as diagonal lines on the grids. (Don't have to be on each of the ten grids.)
Second guess: (9, 0, 0) I used this one because it was the same as the second guess from Level 1. Marked all the possible points again, which gave diagonal lines going the other way, so where they crossed gave some possible hiding places.
Third guess: we took a few goes to work out what was the best point to use for our third guess. We wanted to use another corner of the cube, but our first couple of tries gave us sets of points that we had already got from our first two guesses. We decided on (0, 0, 9) Marked the possible points again, and that gave only one possible hiding place.
Fourth guess: The treasure!
Teachers' Resources
Why do this problem?
This problem offers students a chance to consolidate their understanding of coordinates whilst challenging them to think strategically and work logically.
Possible approach
You may find it useful to print off these grids if your students do not have access to the interactivity.
This problem featured in an NRICH Primary webinar in November 2021, and in an NRICH Primary and Secondary webinar in September 2022.
Demonstrate the Level 1 problem to the class, either using the interactivity or with a grid drawn on the board.
Give students about 10 minutes to work on the problem, either at computers, or on paper in pairs (taking it in turns to choose where the treasure is and give the distances). Pairs can keep score of the number of guesses each student required to find the treasure - the one with the lowest score wins.
Ask the class to share efficient strategies/useful ideas. Encourage the students to consider all the points that satisfy each condition, and to look at the shape of this locus. Re-emphasise that the problem is to develop a strategy to find the treasure with the minimum number of guesses (at Levels 1 and 2, with the appropriate strategy, it is always possible to find the treasure in fewer than four guesses).
Return to the computers/pairs to work on the suggested strategies. Provide squared paper for rough jottings.
If students are familiar with coordinates in four quadrants, the Level 2 game can be an excellent context for practising these. (The level can be changed by clicking on the purple cog in the top right of the interactivity.) Encourage students to do their work on paper.
Key questions
Which points satisfy the conditions given so far?
How can you narrow down the possibilities?
Possible support
Possible extension
The Level 3 game provides an interesting challenge: the searching area is restricted to the pink region, although the treasure may be anywhere on the grid. Users are allowed one 'final answer' guess outside the pink region to locate the treasure.
The Level 4 game provides a challenging context in which to think about 3-dimensional coordinates and it can be explored in 3D Treasure Hunt.
Again, the challenge is to develop a strategy to find the treasure with the minimum number of guesses (at Levels 3 and 4, with the appropriate strategy, it is always possible to find the treasure in fewer than five guesses).