Geometry Puzzle: Cut The Figure to Make 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.