These should do it: Just to show another example:


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.


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 $\...


I think this is the answer.


Of course we need to use Pythagoras. This leads to the following solution: Here is another more compact solution.


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 ...


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).


Work in progress (may not be optimal) Picture: Diagram of contacts: Coordinates (unconstrained circles at reduced precision):


This is a generalisation:


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, ...


The shapes could be red and green in this picture: Strategy: More examples:


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&...


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 ...



On 4x4 chessboard with two pieces, I found by exhaustive search.


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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