# [seqfan] Re: Packing many different square sizes in a square

israel at math.ubc.ca israel at math.ubc.ca
Wed Jun 19 09:16:44 CEST 2013

```Well, I can see why the ceiling(n/2) couldn't continue much past n=16:
squares of side 1,2, ..., m have total area m^3/3 + m^2/2 + m/6, so you
can't pack them into a square with side less than the ceiling of the square
root of that. It's also clear that you can't pack a square of side m and a
square of side m-1 into a square with side less than 2m-1. Those two
conditions give upper bounds for n=1 to 30 of 1, 1, 2, 2, 3, 3, 4, 4, 5, 5,
6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13

Cheers,
Robert Israel

On Jun 18 2013, Allan Wechsler wrote:

>I thought that all the different changes on packing squares into a square
>had been rung, but apparently it is not so.
>
>Take a square of integer side n > 0. Let A(n) be the largest number of
>different sizes that can be incorporated in a tiling of the big square by
>smaller integer squares.
>
>Up to n=16, the answer seems to be ceiling(n/2), but I cannot pack 9
>different square sizes on a 17x17 board.  The sequence I have so far is
>
>1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,8,9,9,9
>
>and if this is right, it's not in OEIS yet. Can any sequence fanatic
>increase any of these numbers?  Or give more terms? (Or better yet, write
>code to search for tilings that maximize the number of square sizes?) I'll
>submit this soon unless somebody can show that I've erred. Thanks in