Solution #16: Mexican Table Mats

Let the number of circles along the top edge of the mat be x and the number of circles along the side of the mat be y. Then the total number of circles in the mat is

xy

and the number of circles around the edge is

	2x + 2y - 4

Since this must be one-half the total number of circles we have

	xy = 4x + 4y - 8
	xy - 4x - 4y + 16 = 16 - 8
	and (x - 4) (y - 4) = 8

Since x and y must be integers, so must (x - 4) and (y - 4) and these must be factors of 8. The only integer factors of 8 are (8 and 1) and (4 and 2), which give {x = 12, y = 5} and {x = 8, y = 6}.

Hence the mat must be 12 by 5 circles or 8 by 6 circles.

Source: Longley-Cook, L. H.
Categories: Reasoning, Algebra

[Top | Problem]