### The Lily Pond

Freddie Frog visits as many of the leaves as he can on the way to see Sammy Snail but only visits each lily leaf once. Which is the best way for him to go?

### Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

### Treasure Island

Write down what you can see at the coordinates of the treasure island map. The words can be used in a special way to find the buried treasure. Can you work out where it is?

# Cops and Robbers

## Cops and Robbers

Can you help the police locate the robber?

The robber is hiding in a modern city, where the roads are all at right angles and equally spaced. Imagine the grid lines represent roads. The robber is hiding at a crossroads.

Input coordinates to help the police search and find the robber with the fewest guesses. The interactivity gives you the shortest distance you'd have to travel (along the roads) to reach the robber.

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

This text is usually replaced by the Flash movie.

### 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 be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

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 robber is and give the distances). Pairs can keep score of the number of guesses each student required to find the robber - 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 robber with the minimum number of guesses (at Levels 1 and 2, with the appropriate strategy, it is always possible to find the robber in less than 4 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 4 quadrants, the Level 2 game can be an excellent context for practising these. 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 extension

The Level 3 game provides an interesting challenge: the searching area is restricted to the pink region, although the robber may be anywhere on the grid. Users are allowed one 'final answer' guess outside the pink region to locate the robber.

The Level 4 game provides a challenging context in which to think about 3-dimensional coordinates.

Again, the challenge is to develop a strategy to find the robber with the minimum number of guesses (at Levels 3 and 4, with the appropriate strategy, it is always possible to find the robber in less than 5 guesses).

### Possible support

Encourage students to draw the grid on squared paper, and colour code points that are possible; looking at the result of each new piece of information.

A version of the 3-dimensional problem that offers more support is available at Lost on Alpha Prime