Packing squares, again

[bonanova]

What is the area of the smallest square that holds five unit squares without overlap?

[dgreening]

I think the smallest square would be a 3 x 3.

If we start with the most compact area to contain 4 unit squares, it would be a 2 x 2 square. The addition of another unit square will force it to add at least one row or column - thus forcing you to a 2x3 configuration. To meet the criteria of the problem, we must expand that to a 3 x 3

There are multiple configurations [many are just reflections of others

# # #

# # -

- - -

or

# # #

# - -

# - -

or

# - #

- # -

# - #

or ...

[harey]

Place four unit squares into the corners of the resulting square and the 5th one into the middle after a 45 degree rotation.

The diagonal of the new square:
d=sqrt(2)+1+sqrt(2)

giving the surface 4.5+2*sqrt(2) (about 7.33).

Edited October 13, 2013 by harey