Problem 3 (Jane's Quilt)

Jane had just finished making the following patchwork quilt

x	x	x	x	x	x
x	x	x	x	x	x
x	x	x	x	x	x
x	x	x	x	x	x
x	x	x	x	x	x

and marveled at the 30 squares in the quilt. Actually, there are more than 30 squares in the quilt, and even more rectangles in the quilt. Can you determine the total number of squares of all sizes in Jane's quilt and the number of rectangles in the quilt ?

Solution

We first count the squares in the quilt. There are various types of squares ranging from single squares we call 1 x 1 squares to 5 x 5 squares. We have

Type of Square	Number of Squares
1 x 1 squares	30 squares
2 x 2 squares	20 squares
3 x 3 squares	12 squares
4 x 4 squares	6 squares
5 x 5 squares	2 squares
	Total 70

For rectangles, we list all rectangles of different sizes below where a 1 x 3 rectangle is considered the same as a 3 x 1 rectangle. If we simply count the rectangles of the various size, we find

Type	# Rectangles	Type	#Rectangles
1 x 1	30	2 x 6	4
1 x 2	49	3 x 3	12
1 x 3	38	3 x 4	17
1 x 4	24	3 x 5	10
1 x 5	16	3 x 6	3
1 x 6	5	4 x 4	6
2 x 2	20	4 x 5	7
2 x 3	31	4 x 6	2
2 x 4	22	5 x 5	2
2 x 5	13	5 x 6	1
		Total	315

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Last modified on Wednesday, March 17, 1999