# Coordinates - February 2005, All Stages

## Problems

### Late Again

##### Stage: 1 Challenge Level:

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

### Making Shapes

##### Stage: 1 Challenge Level:

Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?

### Arrangements

##### Stage: 2 Challenge Level:

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

### Fred the Class Robot

##### Stage: 2 Challenge Level:

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

### Cartesian Isometric

##### Stage: 2 Challenge Level:

The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form?

### Ten Hidden Squares

##### Stage: 2 Challenge Level:

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

### Eight Hidden Squares

##### Stage: 2 and 3 Challenge Level:

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

### Isosceles Triangles

##### Stage: 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### Lost

##### Stage: 3 Challenge Level:

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

### Square Coordinates

##### Stage: 3 Challenge Level:

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

### Lost on Alpha Prime

##### Stage: 4 Challenge Level:

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

### Tracking Points

##### Stage: 4 Challenge Level:

Can you give the coordinates of the vertices of the fifth point in the patterm on this 3D grid?

### Something in Common

##### Stage: 4 Challenge Level:

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

### Exponential Trend

##### Stage: 5 Challenge Level:

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

### Pitchfork

##### Stage: 5 Challenge Level:

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

### Polar Flower

##### Stage: 5 Challenge Level:

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.