Re: help to solve a sequence problem

On Feb 5, 5:49 am, noble_c <nobl...@xxxxxxxxxxx> wrote:
A difficult puzzle,surely! yes,if there are n (n is an positive odd integer) clubs,and each club's length is 1,2,3,...,n repectively.Can arrange them in a rectangle?
I can't solve it.

You need n == 0 or 3 mod 4 so the sum of the lengths is even.

Clearly not possible for n= 3 (unless you count a "rectangle" with two
sides 0)
or 4. For n=7 a square is possible with {1,6}, {2,5},{3,4},{7}.
For n = 8, a square with {1,8},{2,7},{3,6},{4,5}
For n = 11, a rectangle 16 x 17 with {6,10},{1,3,4,9},{5,11},{2,7,8}
For n = 12, a rectangle 19 x 20 with {1,4,5,9},{3,7,10},{2,6,11},
{8,12}
Proceed by induction to get squares whenever n == 0 or 7 mod 8,
rectangles (n^2+n-4)/8 by (n^2+n+4)/8 whenever n == 3 or 4 mod 8,
from here on.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada





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