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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1026.0. "Square packing problem" by TRUCKS::CHANT (Something different) Thu Feb 09 1989 12:15

    This is an old problem, has any one got any references to it or done any
    work on it.


    SQUARE PACKING PROBLEM

    Determine the minimum number of squares that can be fitted into a square
    of side N given the following :

    1) All squares have integer sides. (N is also an integer)

    2) There is at least one unity square.

    Adrian
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1026.1Here's a puzzle exampleAKQJ10::YARBROUGHI prefer PiMon Feb 13 1989 15:5416
I found one reference, which has only this single example: 

Find a set of 11 integral squares that can be assembled into a 13x13 square.

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Squares of primes are generally difficult to break up efficiently.

Squares with composite sides can usually be profitably reduced to a set of
squares whose sides are divisors plus the dissection of one of those 
squares; i.e. 100^2 = 90^2 + 18*10^2 + (number of squares to make up a 
10x10).

If the side N is a power of 2, the square can be broken up into 3log[2]N+1
smaller squares.

Lynn 
1026.2DWOVAX::YOUNGSharing is what Digital does best.Mon Feb 13 1989 16:4016
    To elaborate on what Lynn said in .1, you can set some upper limits
    on the minumum that you are searching for as follows:
    
    If [ N = (2^T) * K ] and (2^T) is the highest power of 2 that divides
    N, : Then we can set some upper limit U(N):
    
(1)	U(N) = 3T + 1			Where N = 2^T
    
(2)	U(N) = 3T + N			Where T > 0
    
(3)	U(N) = N + 3			Where T = 0
    
    These are probably far from optimal, except for (1), but they provide
    a good starting point.
    
    --  Barry