Question

(a) Suppose you have 56 square tiles, each containing a quarter circle that goes from the midpoint of one side to the midpoint of an adjacent side. If we randomly arrange the 56 tiles into a 8 by 7 rectangle, what is the average number of full circles formed?

(b) Suppose you have 64 equilateral triangular tiles, each containing an arc of a circle that goes from the midpoint of one side to the midpoint of an adjacent side. If we randomly arrange the 64 tiles into a larger equilateral triangle, what is the average number of full circles formed?

Answer 1

4 tiles are arranged to form a circle ie 2 on the row and 2 on the column. Similarly 16 tiles can be arranged in such a manner that 8 tiles in a row of two. Here you get 4 circles. You can take 3 sets of 16 tiles among 56 tiles ie altogether 48 tiles will give you 4*3 sets=12 circles. (Here you get 6 rows of tiles each containing 8 tiles. now to complete the rectangle you need to arrange the remaining 8 tiles as the seventh row.

Each side will have 8 +7=15 tiles as shown in the picture.A circle is formed by two tiles. Except one tile in each row. all the tiles can be arranged to form the circle. Hence the average number of circles =1+2+..+7=7*8/2=28