Hey architects, we have an __app__ this week! Please share some solutions you've come up with. Below are a few prompts to get you started!

**Making squares with 2 smaller squares:**

What is the smallest square you can make using both 2x2 and 3x3 squares? (You must use at least one 2x2 square and at least one 3x3 square in your answer!)

Are there any sizes of squares that are impossible to make using 2x2 and 3x3 squares? If so, can you describe which squares are possible and which are impossible? If not, can you explain how you would construct any square?

How would your answers above change if you used 2x2 and 4x4 squares instead? 3x3 and 4x4 squares? Any two sizes of squares?

**Making squares with 3 smaller squares:**

What is the smallest square you can make using 1x1, 2x2, and 3x3 squares?

Are there any sizes of squares that are impossible to make using 1x1, 2x2, and 3x3 squares? If so, can you describe which squares are possible and which are impossible? If not, can you explain how you would construct any square?

How would your answers above change if you used 2x2, 3x3, and 4x4 squares instead? 2x2, 3x3, and 5x5 squares? Any three sizes of squares?

If you come up with other prompts and variations for others to solve, please share!

Any two sizes of squares?If the two squares have sides a and b, respectively, with a < b, The side n of the smallest square that can be subdivided, is n = m.c.m (a, b) + a. From there any square, where n is a multiple of a or b.Ramon found a great solution to the problem of making any square with

2 smaller squares.Ramon's solution states that the smallest square of side-length

that can be built out two smaller squares of side-lengthsnandais found bybn = m.cm. (a,b) + a.As an example ... if we're working with 2x2 and 3x3 squares, a = 2, b =3.

Note that m.c.m (a,b) = the least common multiple of a and b.

n = m.c.m (2,3) + 2

n = 6 + 2

n = 8.

When we explore this on the app, we do confirm that an 8x8 grid can be filled with 2x2s and 3x3s.

Next example explores 3x3 and 4x4 squares:

n = m.c.m(3,4) + 3

n = 12 + 3

n = 15

We can confirm this on the app showing that a 15x15 grid can be filled in with 3x3 and 4x4 squares! It's interesting that both of these examples form a similar structure.

¿Cuál es el cuadrado más pequeño que puedes hacer usando cuadrados 1x1, 2x2 y 3x3?

El cuadrado más pequeño será un cuadrado de 5x5. Para cualesquiera 3 tamaños de cuadrados: a, b y c, si b y c son múltiplos de a, el lado n del cuadrado más pequeño que se puede hacer es n=b+c.