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An activity making various patterns with 2 x 1 rectangular tiles.

Fault-free Rectangles

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Sammy and George from Ardingly, and Bilge, age 12 all sent in correct answers to the first part of the question.

As George explains:

A rectangle doesn't necessarily have to be a oblong it can be a square. All you have to do is build a fault free square which is easy:
You have to put the white square in the middle and then place the red oblongs (so that there are no faults) around the edge of it.

The fault-free rectangle looks like this:


4 red, 1 white

Congratulations to V. Abhijit, age 15 who sent in a fuller solution:

The key behind making a fault free rectangle is this:
Start with a 1 by 1 rectangle, place it at the centre
Now, place 4 2 by 1 rectangles around the 1 by 1 rectangle This will form a fault free rectangle(actually a square, but nevertheless its a rectangle!!)
Thus if you start with a 2 by 1 rectangle, place 2 of them at the centre and then place 4 3 by 1 rectangles around the initial arrangement.

This rectangle is the one made by the red and green rods:
red and green rods

V. Abhijit goes on to generalise...

I noticed that you get a fault free rectangle when I used an "N by 1" rectangle along with an "N+1 by 1" rectangle. I had to put N "N by 1" rectangles at the centre (that forms a square) and then had to place 4 "N+1 by 1" rectangles around the arrangement to get a fault free rectangle(that turned out to be a square)
 

V. Abhijit gives the smallest fault-free rectangle that can be made using 2 by 1 rectangles as that shown below:

 
 
smallest fault-free rectangle
 

Can you find another?