Expert: Chanda Walker - 1/10/2009

Question
If I have a five row by five column quilt (25 squares), how many rectangles are in that quilt?

Answer
If this is a question regarding quilting, then I don't think I have a good answer for you.

If on the other hand this is a math problem, I *can* help you.

There are 25 squares so that starts our sum.

Next, let's count how many rectangles can be made of 2 squares.  We can have 4 "tall" rectangles per column.  We can also have 4 "flat" rectangles per row.  So we have 4*5 + 4*5 or 40 two square rectangles.

Our sum now is up to 25+40 or 65.

Let's look at rectangles made of 3 squares.  We can have 3 "tall" rectangles per column and 3"flat" rectangles per row.  So that is 3*5 + 3*5 = 30.

Our sum now is up to 25+40+30 = 95.

Let's look at rectangles made of 4 squares.  There are two ways to do this.  We can have a 2x2 square or a 4x1 rectangle.  Let's do the 4x1's first.

There are 2 "tall" rectangles per column adn 2 "flat" rectangles per row.  That is 2*5 + 2*5 =20.

As far as 2x2 squares, we'll have to talk about the position of the top left square.  That top left square can be in 4 positions in the top row.  And 4 in the second row, the third row and the fourth row.  But it can't go into the fifth row.  So we have 4*4 places it can go.

Now our sum is 25+40+30+20+16 =131.

Next, rectangles made of 5 squares.  There can be only 1 per column and one per row.  So that is 5+5.

Our sum:25+40+30+20+16+10 =141.

Rectangles made of 6 squares is next.  We can do 2x3's and 3x2's.  I'll use that upper left trick again.  The upper left square of the 2x3 can be in 4 positions in the top three rows.  So there are 4x3 of these.  Likewise for the 3x2's.

25+40+30+20+16+10+24 =165.

Whew, no rectangles made of 7 squares!

Rectangles made of 8 squares can be 4x2's and 2x4's.  That upper left trick again:  The 4x2's can be placed in 2 places per row and in 4 columns.  Likewise for the 2x4's.  So 16 places these can go.


25+40+30+20+16+10+24+16 =181.

9 square rectangles must be 3x3's.  There can be 3 per row in 3 columns.  There are only 9 of these!

25+40+30+20+16+10+24+16+9 =190.

Ten squared rectangles can be made of 5x2's and 2x5's.  There can be 3 "flat" ones and 3 "tall" ones.

25+40+30+20+16+10+24+16+9+6 =196.

No 11 squares.

Twelve squared rectangles must be made of 3x4 or 4x3.  There are 2 per row and 3 per column for "flat" ones and likewise for "tall" ones.  That makes 12 in total.

25+40+30+20+16+10+24+16+9+6+12 =208.

No 13 or 14 squared rectangles.

There are only 6 ways to do 15 squared rectangles.

25+40+30+20+16+10+24+16+9+6+12+6 =214.

There are only 4 ways to do 16 squared rectangles.

25+40+30+20+16+10+24+16+9+6+12+6+4 =220.

No 17,18,19 squared rectangles.

There are exactly 4 20 square rectangles.

25+40+30+20+16+10+24+16+9+6+12+6+4+4 =224.

There aren't any 21,22,23,or24 squared rectangles.

And finally there is 1 25 squared rectangles.

That makes the sum 225.

There you go.  There are 225 possible rectangles (including squares as a "special" type of rectangle.)