Welcome to NRICH.

 
Pentominoes
By Sudha Srinivasan on Monday, June 24, 2002 - 12:22 pm:

Why can't we fit each pentomino onto a shape made with 8 squares ?

Thanks,

Sudha

By Steve Megson on Monday, June 24, 2002 - 01:56 pm:

I think the easiest way to do this is to try to build such a shape from 8 squares and see where it goes wrong. We know that we must be able to fit img1 into our shape, so we'll start from that shape and hope that by adding another 3 squares we can fit all the other pentominoes in.

We have this pentomino:
img2
which means that we must add squares above and below the row of 5 in some column.

We also have
img3
which means that one of the 2 squares we added must have another square next to it for this pentomino to fit onto.

Now we've used up all 8 squares, and get a shape looking something like
img4
But now we have nowhere in this shape to put this pentomino
img5

So any 8-square shape that fits all of the first 3 pentominoes we looked at can not also fit the last one.

Steve

By Sudha Srinivasan on Tuesday, June 25, 2002 - 09:44 am:

Thank you steve!

If all the pentominoes cannot fit in a shape made of 8 squares, what is the minimum number of squares required to achieve this? (I believe it is 9). Is there any proof or explanation as to why this is so?

sudha

By William Astle on Monday, July 01, 2002 - 10:02 pm:

Use Steve's method to look for a shape with 9 squares...