IBM Research | Ponder This | June 2003 solutions

# Ponder This

## June 2003

<<May June July>>

We began by seeking a solution for the least possible value of N, first determining its topology. Clearly two adjacent rectangles cannot have sides that exactly coincide, since then the two equal rectangles would be congruent. Also, we can assume that no rectangle has a side of 1, because then the other rectangles would exhibit an acceptable topology for a smaller N. Exhaustively considering cases, one finds only the following three possible topologies for N <= 7: where the labels refer to lengths of segments of the sides of the rectangles. The requirement of equal areas yields N equations in N+1 unknowns:

(0) 5a = 1/(x+y) , (1) 7a = 1/(w+x+y) , (2) 7a = 1/(w+x+y) ,
5(b+c) = 1/x , 7(b+c) = 1/w , 7b = 1/(w+x) ,
5b = 1/y , 7d = 1/(w+x) , 7(c+d) = 1/w ,
5c = 1/(y+z) , 7b = 1/(x+y) , 7c = 1/x ,
5(a+b) = 1/z ; 7c = 1/x , 7d = 1/(x+y+z) ,
7(c+d) = 1/(y+z) , 7(b+c) = 1/y
7(a+b) = 1/z ; 7(a+b+c) = 1/z .

One may eliminate a, b, c, and (except in case (0)) d from the above to obtain the homogeneous equations

(0) 1/y + 1/(y+z) = 1/x ,
1/(x+y) + 1/y = 1/z ;

(1) 1/(x+y) + 1/x = 1/w ,
1/x + 1/(w+x) = 1/(y+z) ,
1/(w+x+y) + 1/(x+y) = 1/z ;

(2) 1/x + 1/(x+y+z) = 1/w ,
1/(w+x) + 1/x = 1/y ,
1/(w+x+y) + 1/(w+x) + 1/x = 1/z .

Solving first (except in case (0)) for w and then for z and then eliminating these, one finds

(0) z = y(x + y)/(x + 2y) ,
x**2 + xy -y**2 = 0 ;

(1) w = x(x + y)/(2x + y) ,
z = (x + y)(3x + y)/(5x + 2y) ,
y(38x**2 + 38xy + 9y**2) = 0 ;

(2) w = x(2y - x)/(x - y) ,
z = y(2x - y)/(3x - 2y) ,
(x + y)(9x**2 - 20xy + 9y**2) = 0 .

In case (0) it follows that z = x, making some of the rectangles congruent.

The last equation for case (1) means either that y = 0 or that either x or y is negative. All of these alternatives are unacceptable, as is the alternative y = -x for case (2).

So using the second factor in the last equation for case (2), one may solve for y in terms of x and then use that result together with the other 6 independent equations for case (2) and an 8th equation,

w + x + y + z = 1 ,
to find, where Q = +sqrt(19),

w = (8-Q)/15 = 0.242740 , a = (8-Q)/21 = 0.173386 ,
x = (1+Q)/15 = 0.357260 , b = 5/21 = 0.238095 ,
y = (-1+Q)/15 = 0.223927 , c = 5(-1+Q)/42 = 0.399869 ,
z = (7-Q)/15 = 0.176073 , d = (7-Q)/14 = 0.188650 ,

yielding rectangles with (vertical by horizontal) sides

(8+Q)/15 by (8-Q)/21 , 3/5 by 5/21 ,
(8-Q)/15 by (8+Q)/21 , (1+Q)/15 by 5(-1+Q)/42 ,
(7+Q)/15 by (7-Q)/14 , (-1+Q)/15 by 5(1+Q)/42 , and
(7-Q)/15 by (7+Q)/14 .

(The solution with the opposite sign for Q is rejected because then some of these quantities would be negative.) We have established that the least N = 7.

A non-unique solution for each N > 7 can be built up recursively, starting with the unique solution for N = 7. Given a solution for N, a solution for N+1 can be constructed as follows: If N is odd, attach a 1/N by 1 rectangle at either the top or the bottom and then shrink the vertical dimensions of all rectangles by a factor N/(N+1); if N is even, attach a 1 by 1/N rectangle at either the left or the right and then shrink all horizontal dimensions by a factor N/(N+1). Clearly each added rectangle, being longer that any of its predecessors, is congruent to none of them. However, one must also be sure that none of the 7 original rectangles become mutually congruent as they shrink, which is now shown: The cumulative change in the vertical-to-horizontal ratios of the sides of the rectangles is by a rational factor f = (7/8)(9/8)(9/10)(11/10)...(N/(N+1)) or ((N+1)/N), so that 7/8 <= f <= 1. Consequently sides of two distinct original rectangles can become equal only if they start (when N = 7) with a rational ratio r in the range 1 <= r <= 8/7. However, the only rational original ratios are 14/25, 14/15, and 7/5, all of which lie outside this range. Therefore there is a solution for each N >= 7 but none for any N < 7. QED

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