64
These should do it:
Just to show another example:
21
I made folds like this:
These puzzles are getting harder and harder. In this case it was impossible for me to eyeball where any face would be, and it only started to make sense once I started cutting and folding.
20
If a rectangular piece of chessboard is $a\times b$ squares in size, then its diagonal squared is $a^2+b^2$ and its area squared is $a^2\cdot b^2$, and therefore the quantities $a^2,b^2$ are the roots of the quadratic equation $x^2-D^2x+A^2=0$ where $D,A$ are the diagonal and area. But this is enough to determine $\{a^2,b^2\}$ completely: these values are $\...
13
I think this is the answer.
12
Of course we need to use Pythagoras.
This leads to the following solution:
Here is another more compact solution.
11
Improved answer
This is similar to another answer but with a smaller area. I worked it out completely independently, then noticed its similarity. The 3" dish is in a different place, and it is not an adjustment based on that answer. It was generated by a C program I wrote for this purpose. It gave my previous answers and has been spitting out smaller ...
9
The galaxy should be folded like this
This is a crude model I made of the cube, with the joins highlighted (might have missed one).
8
Work in progress (may not be optimal)
Picture:
Diagram of contacts:
Coordinates (unconstrained circles at reduced precision):
8
This is a generalisation:
8
Update: Please read below (under the double underlines) explanation first before coming back here.
I just realized from user65284's answer that we can flip the pieces. Thus, the lowerbound can be increased to:
And here are some illustrations in action:
Here I will give a lowerbound for (original) contiguous case, which is there are at least:
Visually, ...
7
The shapes could be red and green in this picture:
Strategy:
More examples:
6
I used a nonlinear optimization solver, with variables $x_i$, $y_i$, $w$, $h$. The problem is to minimize $w\cdot h$ subject to:
\begin{align}
i \le x_i &\le w - i &&\text{for $i\in\{1,\dots,12\}$}\\
i \le y_i &\le h - i &&\text{for $i\in\{1,\dots,12\}$}\\
(x_i - x_j)^2 + (y_i - y_j)^2 &\ge (i + j)^2 &&\text{for $1\le i&...
6
There is another pair of shapes. I did a complete search but only to size 10 for one piece. The solution above has pieces with area 4 and 10, the second I found has 7 and 7. I show them both. Note that there are four distinct positions for the domino hole, all others are rotations/reflections of these.
For completeness, I show the 14 ways of doing this with ...
4
3
On 4x4 chessboard with two pieces, I found
by exhaustive search.
1
Building on the answers by Elias and CiaPan... Elias gave this "binary search"-based answer.
But it's interesting to notice that we can also do it with this "unbalanced binary search":
Or even like these:
Or the two solutions with the red piece on the outside, which I'm too lazy to draw out.
I don't know if there are other solutions. I actually asked ...
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