Florian F's 2nd pattern is far and away my favorite, but if anyone was curious, I'll post my answers.
First I wanted to show an example of something that doesn't work but really seems like it should:
It's just like the third example from the question, but it uses joinery such that the pieces slide together at an angle. It comes apart in the same way, ...
To a real-life problem I had to give a real-life answer:
But you asked for an actual tiling, without gaps, so here it is.
PS: there is a simpler pattern where pairs disassemble with a single translation:
Maybe this works?
Note: I am using MS Excel to finish this. The grids are not perfect squares, so they, when analysed using this image, may not be accurately identical. Yet I hope you all get the idea. Thanks!
No. The rectangle has an area of $75$ square units. The only other rectangle with integer sides that would have the same area would be $3*25$ (or $1*75$ technically, I guess).
The rectangles you have are $3*6, 2*3, 3*2, 2*4, 4*2, 10*2$ and $9*1$. The $9*1$ needs a $9*2$ paired with it to fill those rows and prevent leaving a $1*x$ space that cannot be ...
If the triangle has three weights, we can subtract the smallest value from each side. That reduces to the case of only two unknown weights and a zero in the third pan.
The original problem defines the maximum weight as 40. Rather than examine this exact case, I assume an arbitrarily large MAX weight and solved for that. This will give an ...
Suppose we label the corner on the table like this:
Now we want to move from $A$ to $D$.
Now, imagine the table like this:
Now, to hits all $4$ edges, that means
To get the shortest path,
Which is like this: