|
The only sets of dimensions for which over half the area is within two inches of the edge, but if either dimension is increased by 1, that that is no longer true, are:
multiple pieces
used
H W H W single double triple
9 39 13 3 39 156 351
10 23 23 10 230 920 2070
11 18 18 11 198 792 1782
12 15 5 4 20 80 180
13 14 14 13 182 728 1638
14 13 13 14 182 728 1638
15 12 4 5 20 80 180
18 11 11 18 198 792 1782
23 10 10 23 230 920 2070
39 9 3 13 39 156 351
By way of explanation: the multiple used columns represent how many rows and columns would have to be used of repetitions of the given page size to form a square map, so for example, in the first row, 13 rows of 9-inch-high maps in 3 columns 39 inches wide would make the smallest possible square map using that shape of piece, where the square map would be 117 inches square. That arrangement has 39 pieces, but if you wanted to double each dimension of the large map that's fit together, it would have 156 pieces. If you tripled each dimension it would require 351 pieces.
So these are the only numbers of pieces that would result in a square overall map that had been cut up. Of these, only one lies between 50 and 150, and that is 80, which results from pieces that are 12 inches by 15 inches, which is thus the answer. The 80 pieces results from having used twice the 5*4 repetition of the dimensions, so it used 10 of the 12-inch dimension and 8 of the 15-inch dimension, and the original square map was 120 inches by 120 inches, or 10 feet by 10 feet.
From Enigma No. 1623, "Over the edge", by Susan Denham, New Scientist, 27 November 2010, page 28.
|
