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# A square region with perimeter 60 inches is made with square inch tiles. Bob removes one tile from the square and rearranges the remaining t

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A square region with perimeter 60 inches is made with square inch tiles. Bob removes one tile from the square and rearranges the remaining tiles without any overlap to make a rectangular region with minimum perimeter. How many inches are in the perimeter?

Aug 5, 2017

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If the original perimeter was 60 inches, there must have been 15 tiles per side and 15 * 15  tiles = 225 tiles in all for an area of 225 in^2

Removing one tile will give us an area of 224 in^2  with 224 tiles

It can be shown that the perimeter will be minimized whenever  the difference between the length and width is  minimized

The divisors of 224  are  :  1 | 2 | 4 | 7 | 8 | 14 | 16 | 28 | 32 | 56 | 112 | 224

And the possible lengths and widhts giving and area of 224 in^2  with the associated perimeters, P, is as follows :

1 , 224    P  =  450

2 , 112    P  =  228

4 ,  56     P =  120

7,   32     P  =   78

8,   28     P  =   72

14, 16     P =    60

So......14  x 16   tiles   minimizes the perimeter =  60 inches   [ oddly..... the same perimeter before we removed a tile  !!! ]

Note......this will always happen when we remove exactly one tile from any square as in this situation..... if s is the side of the square....4s is the perimeter.....if we remove one tile,  the new length and width that minimizes the perimeter is

(s - 1), (s +1)....but the perimeter is also  2 [(s - 1) + (s +1)]  =  2 [ 2s ]  = 4s   !!!!

Aug 5, 2017
edited by CPhill  Aug 5, 2017
edited by CPhill  Aug 5, 2017