# [seqfan] Re: Two tenuously related problems concerning squares

israel at math.ubc.ca israel at math.ubc.ca
Sun Jun 16 19:31:07 CEST 2013

```1) It is always possible, and not just for squares.  A key-phrase to look
up is "Zeckendorf representation".

Cheers,
Robert Israel

n Jun 16 2013, Alonso Del Arte wrote:

>1. Is it always possible to express n^2 as a sum of n distinct Fibonacci
>numbers, or at least almost so but allowing two 1s? e.g.,
>
> 1 = 1
> 4 = 1 + 3
> 9 = 1 + 3 + 5
>16 = 1 + 2 + 5 + 8
>25 = 1 + 1 + 2 + 8 + 13
>
>2. At the time Howard Eves wrote *Mathematical Reminiscences*, a division
>of a 175 x 175 square into 24 smaller squares, none of them equal, was the
>record. The interest so far appears to be in using as few smaller squares
>as possible. But what I'm curious about is: what is the minimum possible
>number of equal squares? For n = 175, that would be 0. But for say, n = 3,
>that would be 5, since the 3 x 3 square can be divided into a single 2 x 2
>and five 1 x 1s.
>
>Al
>
>
```