One of the things that we discussed quite a lot in the facilitator meeting was the difference between algebraic and geometric "constraints". By algebraic, we are referring to the fact that the sum of the areas of the squares needs to equal a perfect square as well.

For instance, if we are trying to construct a 7x7 grid with 3x3 and 2x2 tiles, we know that we only one or five 3x3s will allow us to have an area remaining that will be a multiple of 4 (meaning that 2x2s can fit). If there is one 3x3 we can have ten 2x2s and if there are five 3x3s we can have one 2x2.

You actually notice that we can't even fit five 3x3s no matter how hard we try... so only one is allowed.

However, now we get into some interesting geometric constraints having to do with the edges of the board. If the edge length is 7, the only way we can construct this with 3s and 2s is one 3 and two 2s, meaning that one 3x3 needs to lie on every edge of the board and two 2x2s need to lie on every edge of the board, like below:

We see however that this yields contradictions with our algebraic constraints. How can we have every side look like this or some variation of this with only having one 3x3 on the entire board? We can't!

Using these types of constraints is a quick way to prove whether you will or will not be able to fill in certain sizes. This problem was super fun and I can't wait for next week's webinar!

This puzzle and the corresponding questions are very interesting! It involves the use of algebra to find possibilities but also requires you to think about whether or not fitting those possibilities in an n x n grid is geometrically possible.

eliasgitterman1

Jul 31, 2020

I really enjoyed this puzzle. Many kids really got into it and learned to how to decode it.

I agree this is an interesting problem, I was wondering if any group was able to solve it? That is, get a set of rules that would allow us to find all the possible solutions for a given combination of small square sizes and large square (or rectangle) to be covered? Even a general answer to the first part - what combinations of small square sizes are possible to make the correct total area (even if the geometry wouldn't let it be a solution) - the algebraic constraint - would be interesting.

A rather simple Strategy, put the biggest one in a corner, then the next biggest on edges of that

What if the goal were to use the fewest number of squares possible? Does your method work with this new goal?

You can use the app and share screenshots of your work!

One of the things that we discussed quite a lot in the facilitator meeting was the difference between algebraic and geometric "constraints". By algebraic, we are referring to the fact that the sum of the areas of the squares needs to equal a perfect square as well.

For instance, if we are trying to construct a 7x7 grid with 3x3 and 2x2 tiles, we know that we only one or five 3x3s will allow us to have an area remaining that will be a multiple of 4 (meaning that 2x2s can fit). If there is one 3x3 we can have ten 2x2s and if there are five 3x3s we can have one 2x2.

You actually notice that we can't even fit five 3x3s no matter how hard we try... so only one is allowed.

However, now we get into some interesting geometric constraints having to do with the edges of the board. If the edge length is 7, the only way we can construct this with 3s and 2s is one 3 and two 2s, meaning that one 3x3 needs to lie on every edge of the board and two 2x2s need to lie on every edge of the board, like below:

We see however that this yields contradictions with our algebraic constraints. How can we have every side look like this or some variation of this with only having one 3x3 on the entire board? We can't!

Using these types of constraints is a quick way to prove whether you will or will not be able to fill in certain sizes. This problem was super fun and I can't wait for next week's webinar!

This is a wonderful way to describe these two constraints! Thank you for sharing.

This puzzle and the corresponding questions are very interesting! It involves the use of algebra to find possibilities but also requires you to think about whether or not fitting those possibilities in an n x n grid is geometrically possible.

I really enjoyed this puzzle. Many kids really got into it and learned to how to decode it.

The puzzle and questions were very engaging.

I agree this is an interesting problem, I was wondering if any group was able to solve it? That is, get a set of rules that would allow us to find all the possible solutions for a given combination of small square sizes and large square (or rectangle) to be covered? Even a general answer to the first part - what combinations of small square sizes are possible to make the correct total area (even if the geometry wouldn't let it be a solution) - the algebraic constraint - would be interesting.

Hey Stan,

Ramon and I explored the first exploration answering the question: making any square with

2 smaller squares.https://www.jrmfactivities.org/forum/squareland-architect/explorations-1Perhaps you can help us explore with

3 smaller squares.