[seqfan] Re: Two tenuously related problems concerning squares

israel at math.ubc.ca israel at math.ubc.ca
Sun Jun 16 22:40:43 CEST 2013


Oops, sorry, somehow I seem to have missed the second "n".  Please ignore.

Robert Israel

On Jun 16 2013, israel at math.ubc.ca wrote:

>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
>>
>>
>
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