Cutting a figure into two identical parts

[bob012345]

Here is a little puzzle from Pierre Berloquin’s book 100 Geometric Games. The goal is to cut the figure once to make two identical parts. Have fun!

 

[kuruman]

Does "identical" mean that after the object is cut into two, one can superimpose the two pieces into one with no more than a simple rotation in the plane of the screen?

 

[bob012345]

Does "identical" mean that after the object is cut into two, one can superimpose the two pieces into one with no more than a simple rotation in the plane of the screen?

Identical here means they have the same shape. Think of it like two pieces of physical paper. Only the shapes matter.

 

[kuruman]

Identical here means they have the same shape. Think of it like two pieces of physical paper. Only the shapes matter.

This doesn't help. A cut along the red line produces two pieces that have the same shape, namely rectangles. Is this a valid solution? If not, why not?

 

[FactChecker]

Identical here means they have the same shape. Think of it like two pieces of physical paper. Only the shapes matter.

Can you flip one, so that a clockwise outer perimeter becomes a counter-clockwise perimeter?

ADDED: Sorry. The one I thought of with a "flip" does not solve it and the real solution does not require a flip.

 

[bob012345]

Can you flip one, so that a clockwise outer perimeter becomes a counter-clockwise perimeter?

Yes.

 

[bob012345]

View attachment 372339This doesn't help. A cut along the red line produces two pieces that have the same shape, namely rectangles. Is this a valid solution? If not, why not?

Same shape in this context means that they overlap, they are the same size and thus are identical so that is not a valid solution as I understand the problem.

 

[bob012345]

Spoiler

Here is one solution. There is at least one more.

[\SPOILER]

 

[haruspex]

Here is a little puzzle from Pierre Berloquin’s book 100 Geometric Games. The goal is to cut the figure once to make two identical parts. Have fun!

View attachment 372336

The original text does not say "once". You added that in your instruction. But what does cutting once mean? It suggests a single straight cut, but that appears impossible. Do you mean one continuous line, but not necessarily straight? I can do that, but it seems too easy.

 

[bob012345]

The original text does not say "once". You added that in your instruction. But what does cutting once mean? It suggests a single straight cut, but that appears impossible. Do you mean one continuous line, but not necessarily straight? I can do that, but it seems too easy.

Yes.

 

[kuruman]

Yes.

OK, then if turns are allowed, it is easy as @haruspex noted. Referring to the figure in post #4, separate in half each of the rectangles with vertical cuts to the red line and then connect the vertical cuts with a horizontal cut.

 

[bob012345]

OK, then if turns are allowed, it is easy as @haruspex noted. Referring to the figure in post #4, separate in half each of the rectangles with vertical cuts to the red line and then connect the vertical cuts with a horizontal cut.

That gives what I showed in post #8. But there is at least one other solution. Who can find it and possibly more?

 

[bob012345]

BTW, if we label the bottom left corner of the original figure as (0,0) and the upper right corner as (5,3), we can describe the cut above as $$(3,3)->(3,2)->(2,2)->(2,0)$$ and we can use that method to describe other cuts.

 

[bob012345]

Well, it’s been 2 days and no one has offered up a different solution than the one above by @kuruman which I believe is the simplest solution. There exist two other solutions. Interestingly, the solution given so far is not the ‘book’ solution! I’ll give the book solution now so perhaps that will stimulate finding the third solution. Setting two small squares as one unit, the lower left corner is $(0,0)$ and the upper right corner is $(5,3)$ make the cut $$(2,3)->(2,1)->(3,1)->(3,0)$$

Then flip one of the two parts.

 

[symbolipoint]

I'm trying it mentally. Without rechecking mechanically, I 'see' to cut along vertical segment number 4. It seems to work. Certainly laziness is stopping me from trying with a physical model.

 

[bob012345]

I'm trying it mentally. Without rechecking mechanically, I 'see' to cut along vertical segment number 4. It seems to work. Certainly laziness is stopping me from trying with a physical model.

Can you describe the cut using the coordinate system I suggested above?

 

[symbolipoint]

Can you describe the cut using the coordinate system I suggested above?

If I try, really hard, but I am not going to take the trouble.

 

[symbolipoint]

I constructed a physical paper model. What I said in post #15 was wrong. I can't do this puzzle.

 

[bob012345]

I constructed a physical paper model. What I said in post #15 was wrong. I can't do this puzzle.

Before I just give the answer, I offer a clue in the form of a riddle. The key to unlocking this mystery is persistence.

 

[kuruman]

Is this the third solution you had in mind?

Spoiler

Two Intertwined letters F

To get the mass symmetry, each piece must have 24 squares which I assumed to be six 4x4 squares. Here is one way to assemble them symmetrically.

 

[bob012345]

Is this the third solution you had in mind?

Spoiler

Two Intertwined letters F
View attachment 372437To get the mass symmetry, each piece must have 24 squares which I assumed to be six 4x4 squares. Here is one way to assemble them symmetrically.

You nailed it! Did the clue help?

 

[kuruman]

You nailed it! Did the clue help?

I understood its importance after solving it. A key is a key is a key.

I looked for other solutions, but I did not find any, at least with 4x4 squares.