what should be my hypothesis for this LEGO project?

[nmehta]

LEGO BRICKS (1X1, 1X2, 1X3, 1X4, 2X2, 2X3, 2X4) HOW MANY TOTAL COMBINATION'S I CAN MAKE AND WILL THAT BE 'ODD' OR 'EVEN'. I WILL KEEP ONE LEGO BRICK FIXED AS BASE AND OTHER WILL BE PLACE IN DIFFERENT WAYS. TO START WITH I WILL USE TWO SAME COLOR BRICKS. SO FAR WE ALL KNOW ODD + ODD = EVEN, EVEN + ODD = EVEN, BUT WHEN WE USE 2X2 VS. 2X2 BRICKS WE GET 9 COMBINATION AND THAT IS 'ODD'. I.E FROM TWO EVEN BRICKS
WE GET 'ODD'. WHAT ABOUT OTHER POSSIBILITIES?

[nmehta]

I tried to work my son on the attached. Q.1 - are we on right direction here? Q.2 - what do we need to think differently here?

[hhemken]

nmehta,

Please forgive my slowness, but I would need more information with regards to what "odd" and "even" mean, and what the valid stacking operations are. Must the smaller brick always be completely contained by the larger one, for example? People who review this project will very likely want to see photographs of the combinations, and possibly diagrams showing all possible orientations for each pair.

Thanks!

[nmehta]

Hello hhemken,

please see attached as asked by you.

[nmehta]

Hello hhemken,

please see attached as asked by you.

Thanks for taking time and interest here.

Sincerely,

[gbrebner]

It looks like the basic question is: given two blocks, what is the number of ways of putting one on top of the other with at least one overlap?

Now, for some (small) given set of blocks, one can figure this out, just by counting. This looks like what you've been doing so far. More mathematically satisfying would be to come up with some formula for this number, given some representation of two blocks. This is actually pretty hard, getting you into the area of combinatorics. This can be graduate school stuff, certainly not first grade.

I'm assuming that's why you've simplified the question, just to get a simple formula for predicting whether the count is odd or even. In other words, is it divisible by two? My guess would be that the formula for the actual count is sufficiently complicated that the number for any pair of blocks would essentially be "random" in terms of whether the count is divisible by two. So there isn't going to be an easy way here.

Potentially there's an interesting project here in terms of seeing how numbers of possibilities can increase very rapidly. Rather than looking at oddness or evenness, you could look at how the number of possibilities grows as the two blocks get larger. You'll see an exponential growth, for sure. This could be plotted on a graph.

Gordon.

[nmehta]

Hello Gordon,

You have interpreted 100% correct. I like your recommendation here. It seems more meaningful to do by a first grader but the only question will be to set scientific hypothesis here - any suggestions here? Also there will be some work on presentation part here but i want to see how my son comes up with his idea - to draw or to keep actual LEGO?

Sincerely,

Niyant

[gbrebner]

Actually, I found myself thinking about this again during the night. Assuming I understand your set-up correctly, then it's not so fearsome a combinatorics exercise as I first thought. Not first grade math, but I derived a formula for the count. It grows quadratically rather than exponentially in fact. The good news is that the odd/even question fell out too. So I suggest that the following is the case.

If the base block is of size a x b then the count will be odd if and only if:
(i) when the other block is square, size c x c: a+c and b+c are both even;
(ii) when the other block is non-square, size c x d: a+b and c+d are both odd.

No guarantees on my math, but maybe worth trying as a hypothesis!

Gordon.