Someone took a huge square map and cut it into identical rectangular pieces by cutting straight down even columns and across even rows, thus preserving the orientation (portrait or landscape mode) of each piece identically, and then bound the pieces together to make an atlas, with each piece becoming a page. Each piece was an integral number of inches both vertically and horizontally.

There were between 50 and 150 pieces altogether, and it turned out that on each page, over 50% of the area was within two inches of the edge of the paper. If either dimension (height or width) of the page had been just one inch larger it would have been no longer true that over 50% of the page was within two inches of the edge.

What was the size of each piece that became a page?

Start with the second part, and a single sheet, area xy. Assuming that exactly half the area were within 2 inches of the edge; then each of x and y, less 2 at each end, and multiplied together, would be half the area;  xy=2(x-4)(y-4), but the centre is a bit smaller than that, so 2(x-4)(y-4)<xy. Then for the augmented sides we need 2((x+1)-4)(y-4)>=(x+1)y, and 2(x-4)((y+1)-4)>=x(y+1) which solve for {9,39}{10,23}{11,18}{12,15}{13,14} and their counterparts with {x,y} reversed.

For the first part; call the huge square n^2. Speaking in general terms,n must be divisible by each of x and y exactly, by reason of the cutting; ax=by=n; and since x and y are different, a=y and b=x: so n=xy. The number of sheets, s, is between 50 and 150: s(xy)=(xy)^2, so s also=xy.

However, none of the potential solutions has xy less than 150; the trick is to find common factors in {x,y} to reduce s. For example, if {x,y}={12,15} then 20 pages make a square sheet, 60*60, and 4 such sheets, 120*120, qualify as 80 pages. If {9,39} then 39 pages make a square sheet, 3*13, but that is less than 50; while 4 such sheets make 156 pages, which is more than 150. None of the other potential solutions have useful common factors.

So the pages must be 12*15, in either order.

Posted by broll on 2011-02-22 14:56:05