### Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Beelines

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

# Lost on Alpha Prime

##### Stage: 4 Challenge Level:

Well done to Trevor from Riccarton High School, and also to Erin, for dealing with each of the three coordinates one at a time.
Keeping two coordinates fixed, while changing the third coordinate and noticing whether the distance to the target metagiff increased or decreased, then stopping when no further decrease is possible, certainly locates the target in the end.

Once we have a method that works the next challenge is to locate the target using the minimum number of guesses.
Michael used algebra to do this.

I needed to draw a diagram to follow his algebra, here is the diagram I drew.

If the metagiff is at point $M$ with coordinates $a$, $b$, and $c$ in the $x$, $y$, and $z$ directions respectively then the difference between the result for $(0,0,0)$ and the result for $(0,0,9)$ will depend only on $c$ because $a$ and $b$ are included in the total distance for both results .

For example if $c$ was $7$, $R(0,0,0)$ would be $a+b+7$ and $R(0,0,9)$ would be $a+b+(9-7)$, or rather $a+b+2$;

In general $c$ is half of ($R(0,0,0) - R(0,0,9) + 9$)

The same logic can be used to find $a$ and $b$:

$a$ is half of $(R(0,0,0) - R(9,0,0) + 9)$

$b$ is half of $(R(0,0,0) - R(0,9,0) + 9)$

So the four results $R(0,0,0)$, $R(9,0,0)$, $R(0,9,0)$ and $R(0,0,9)$ are enough to locate the target.

Do you think you could find the metagiff in less than 4 moves?
If so, how, and will it work in all cases?