# Tiling the Rectangle into Smallest Number of Squares of any Dimension

What is the smallest number of squares of any size we can dissect a rectangle of a given dimension? This was our challenge this week as we explored the strategy of breaking up rectangles into the largest squares possible in order to minimize the number of squares.

Of course the maximum squares with whole number dimension would be the product of the original rectangle’s dimensions. For example, a 3x4 rectangle would allow 12 squares of 1x1 each. But the minimum number of squares would be one 3x3 square and three 1x1 squares remaining for a total 4 square dissection.

When we looked at squares with dimension of consecutive Fibonacci numbers (1 2 3 5 8 13 21 34, etc.), we are then able to create the golden spiral by creating a quarter circle in each square in the dissection (please see the attached pdf on pages 6 and 7).

Page 8 of the pdf have non-rectangular composite figures that are quite challenging forcing the mathematician to not always use the largest square possible. It took me several tries to optimize the smallest number of squares.

Finally, I challenged the older children to dissect rectangles that could not be sketched such as 5 x 102 where we use short division to solve this challenge. First divide 102 by 5 to get 20 5x5 squares with a remainder of 2, then divide 5 by 2 to get two 2x2 squares and finally divide 2 by 1 to get the two remaining 1x1 squares for a total of 24 squares.

What if the dimensions are 11 x 240? First we divide 240 by 11 and then divide 11 by that remainder, and repeat until the remainder is 0, the sum of the non-remainder quotients are the total number of squares. In this case, 28 total squares.

Try these rectangles:

10 x 33

5 x 44

7 x 85

11 x 130

15 x 186

18 x 380

21 x 585

31 x 657

50 x 1111

AttachmentSize
Tiling_the_Rectangle_with_Fewest_Squares.pdf3.14 MB