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## 'Something in Common' printed from http://nrich.maths.org/

For this question you can only draw lines between points on the
grid with integer coordinates, such as (6,-8,-2) or (3,7,0). You
cannot draw lines between points that do not have integer
coordinates e.g. (2.5,1,8) .

It is possible to draw a square of area 2 sq units on a coordinate
grid so that two adjacent vertices are at the points (0,0) and
(1,1) - see diagram below. In fact there are two such squares with
sides $ \surd 2$ and area 2 square units - as shown.

The

Tilted Squares problem also investigates other squares you can
draw by tilting the first side by different amounts. Here is a
square with side: $ \surd 13$ and area 13 sq units.

It is not possible to draw a square of area 3 sq. units on the
grid. Try some squares for yourself and then explain why.

But is it possible to draw a square of area 3 sq. units in a 3D
grid? First, you need to be able to make a side of length $ \surd 3
$. The line joining (0,0,0) to (1,1,1) has a length of $\surd 3$.
How do I know?

Then you need three more sides all the same length that meet at
the vertices and are at right angles to each adjacent side.

How many squares of area 3 square units can you find with this side
in common, and what are the coordinates of their other
vertices?