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Offline Hermit
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#960
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No chance.

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By connecting parallelograms to each other along their sides, you can only obtain a figure with an even number of angles.
Всё не так плохо как Вы думаете. Всё намного хуже!
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Online now Злобный Ящур
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#961
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Why?

Ничто не возникает из ничего и ничто не пропадает бесследно.
Если где-то чего-то убудет, то в другом месте добавится.
(Закон сохранения).

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Offline Hermit
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#963
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Evil Lizard
Why?
Well, I think it's because when you attach a parallelogram to any polygon, it either won't change the number of angles, or it will decrease them by two, or decrease them by four, or increase them by two, or increase them by four. This is easy to show in a diagram. A parallelogram has 4 angles, so when attaching other parallelograms to it, we will always get a figure with an even number of angles. I think there's something about topology here, but I don't know topology.
Всё не так плохо как Вы думаете. Всё намного хуже!
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Online now Злобный Ящур
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#964
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Hermit
I think there's something related to topology here.
Well, you guys are really something... (c) :eek::eek::eek:

Added 2 minutes later
Ment
What did Kosha write? Maybe it's true? You never know...
.

Ничто не возникает из ничего и ничто не пропадает бесследно.
Если где-то чего-то убудет, то в другом месте добавится.
(Закон сохранения).

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Offline Hermit
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#965
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Actually, you can get a shape with an odd number of angles, so you can even get a 111-sided polygon, but it's unlikely to be any shape, because, for example, you can't get a triangle.

Added 6 minutes later
Although, you can't get a convex 111-sided polygon.

Added 8 hours 57 minutes later
I think you can build a maximum of a convex hexagon from parallelograms.
Всё не так плохо как Вы думаете. Всё намного хуже!
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Offline Ment
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#966
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When parallelograms are joined to form a figure, we will have parallel sides, right? Moreover, the sides will be parallel in pairs. Because, for any side of a parallelogram, the following is true: either it is parallel to the opposite side, and both are external sides of the polygon; or one of them is not an external side, but it is adjacent to a side of another parallelogram, so, in a chain, "parallelism" is passed on to the external side.
In a 111-sided polygon, there is an odd number of sides, as a result, at least one side will not be parallel to any other.
 

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Offline Hermit
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#967
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Example of an 111-sided polygon consisting of parallelograms. However, it's definitely not convex :(
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Всё не так плохо как Вы думаете. Всё намного хуже!
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Online now Злобный Ящур
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#971
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Ment
When parallelograms are joined to form a figure, we will have parallel sides, right? Moreover, the sides will be pairwise parallel. Because, for any side of a parallelogram, the following is true: either it is parallel to the opposite side, and both are external sides of the polygon; or one of them is not an external side, but it is adjacent to a side of another parallelogram, so the "parallelism" is passed on to the external side.
In a 111-sided polygon, there is an odd number of sides, so at least one side will not be parallel to any other.
"Two lines parallel to a third line are parallel to each other." That's correct. What kind of topology is this??? 8th-grade geometry is enough...

Added 7 minutes later
Ment
Oh, you can attach several parallelograms to one side...
We also need to consider convexity.

Ничто не возникает из ничего и ничто не пропадает бесследно.
Если где-то чего-то убудет, то в другом месте добавится.
(Закон сохранения).

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Offline Ment
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#972
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We also need to consider convexity.
You can consider this a hint... Well, it seems that to create a convex figure, we cannot attach multiple parallelograms to one side. More precisely, we can, but it will be the same as one parallelogram divided in two (or three/four parts, etc.).
Why won't it be convex? Because there will be two parallel lines (two sides of the parallelograms, opposite to the sides lying on a common line) and a line intersecting them (the common side of the parallelograms). If you calculate the angles, one will definitely be greater than 180 degrees.
It sounds a bit awkward without a drawing, but I don't think anything is missing.
 

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Offline Ment
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#973
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I was told that I am setting a puzzle. Okay. It's a fairly standard puzzle: you need to swap the numbers 4 and 5. Obviously, the numbers can only be moved to an empty cell, and the cell where the number was before becomes empty. But I don't need the solution to this problem. I am interested in the minimum number of moves required to solve this puzzle. A proof is desirable, but I think I won't require it (unless those who answer start solving the puzzle by brute force, then a proof will be needed).
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