Tiling Puzzle
There is a new company, Tile Carpets, that sells carpets as individual tiles that can be laid down side by side to form a single solid plane. These pieces of carpet are sold in sets, where every piece in the set is the same shape and size.
In order to form a single carpet, customers arrange the carpet tiles on the floor with no gaps in between them. This is possible for some shapes, but not for others. Which of the following shapes is impossible to lay down with no gaps?
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10 solutions
or we just use simple logic. Squares are the regular shape of tiles. Squares eliminated. Equilateral triangles are simply 1/2 of a parallelogram, triangles eliminated. They use hexagons on brick tiles, hexagon eliminated. So we are left with pentagon tiles which I have never seen before. :D And well, the math confirmed!
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Moreover, If you like football you will know that pentagons will make the surface 3D, not a plane.
Yes even i thought on the similar terms.Calculation free deduction.
This is just experience, you can't understand why that happens thinking like that
that's much clearer
i have done this :D
The explanation of Thompson answer is this: When you join maximum vertices that can be joined of same shape at a single point ,then the sum of all the angles should be 360 degrees , which in case of pentagon is only 324 degrees(108 X 3) and some space will be remain unused.
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how u can write 180*3 ???
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Pentagon's internal angle is of 108 degrees , so at one point only 3 pentagons can be joined without overlapping . But since 108*3 = 324 (not 360) , some space will left blank and therefore not an answer ;)
Consequently, the equilateral triangle, square, and regular hexagon are also the only regular polygons that can tesselate a plane.
Although this is not a mathematical solution, this is a curious way of predicting that the pentagon is the correct answer:
The square is simply the first one that one eliminates immediately. After that, note that we have two options, the hexagon, and the equilateral triangle... But hey! The hexagon can be divided into 6 equilateral triangles! What that means? That the answer can't be one of them, because in case that the equilateral triangle isn't possible, then the hexagon wouldn't be possible! And there's only one correct answer! So, finally, we're left with the pentagon.
I think the wording of the question is a bit loose. You can of course make triangle tile that can be used to make the pentagon.
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It mentions equilateral triangle, which cannot be joined to make a pentagon.
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No, I am not confused by the analysis I am pointing out that the question as worded is ambiguous. I could be asking the typical tiling question but as worded could be asking another question, ' which of these shapes cannot be made from several congruous tiles. It just needs to be reworded. Sorry I opened this can of worms
thank you for this solution... now i know... well i just follow my observations, without knowing the proper solutions.
if we can make a plane with regular pentagon,then the football would not have hexagons,it would have only pentagons
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Not really a good argument because the football is not a planar surface (so non-Euclidean geometry applies). For example the sum of the angles of a spherical triangle is not 180deg. Tesselation of a spherical surface is an interesting question, but it is different from the one above.
i had just guessed with no real reasoning. Thanks Thompson for the detailed explanation.
thank for this information
If some tiles of same shape are combined together and makes a plane then sum of angles between two adjacent edges of every tile at point of joint must be 360 deg. But in case of pentagon , you try to combine three pentagon then it is not possible because
angle between two adjacent edges=108
suppose 3 pentagon are combined then at the point of joint, sum of angle =3*108=324 not equal to 360 .
Thank you for sharing the explanation. This method can be used to understand tiling even for larger polygons where we can't comfortably just use logic and eliminate options.
There is an interesting reason I knew the answer, not just the mathematical logic. There is a theory that the Pentagon is a pentagon so 'alien invaders' may not surround it with their ships.
nyc
is this a formula to find interior angle? if so give me the general form..... and more briefing will be welcomed... thank u .....
Wao very nice ans
Well put!!
simple logic. not maths
we just use simple logic. Squares are the regular shape of tiles. Squares eliminated. Equilateral triangles are simply 1/2 of a parallelogram, triangles eliminated. They use hexagons on brick tiles, hexagon eliminated. So we are left with pentagon tiles which I have never seen before. :D And well, the math confirmed! in football you will know that pentagons will make the surface 3D, not a plane
All sides are the same length (congruent) and all interior angles are the same size (congruent) for a hexagon. each angle measures up to 720 degrees and there are six sides =>720/6=120 degrees. this property makes it to fit without no gaps
triangle and rectangle can be easily joined without any gaps. we are left with pentagon and hexagon.hexagonal structures are found in bee hives which are tightly packed, so the only valid answer is PENTAGON
Omg. So easy.?
Thanks for this solution...
thank u for the easiest solution
the interior angle must be a factor of 360
triangles only sum up to 1 8 0 ∘ .
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And 1 8 0 is a factor of 3 6 0 since 3 6 0 ÷ 1 8 0 = 2 .
reg pentagon is not comp. symmetric
wonderful
Well, I only used a little reasoning and found out that squares,equilateral triangles and hexagons can perfectly be adjacent in a given space without gaps between them, so, regular pentagon was the only odd one left... Ans. Is regular pentagon...
Simple, just apply tesselation. - Triangles can always be arranged. 1. 360/angle of a square(90)= 4 a whole number 2. 360/angle of a pentagon(108)= ~3.33 not a whole number 3. 360/angle of a hexagon(120)= 3 a whole number
Since angles of a point is 360 degrees, pentagon can't be tessellated cause (angle of pentagon)n = 360. The other shapes can. Therefore, pentagon can't be used
Triangles could be made into à single carpet by keeping them alternately upside down. Square too can be kept side by side without any gaps. For hexagon, I remembered the honey storage which is small hexagons put together without gaps in between. Pentagon is the only figure that couldn't be made into a carpet without gaps.
You can't make an infinitely covered space with pentagons because no matter how you arrange 2 adjacent pentagons, you'll always have an angle of 1 4 4 ∘ between them. With another pentagon, you only fill 1 0 8 ∘ of that space, with a remaining 3 6 ∘ , which is impossible to fill with another pentagon.
because of five sides of regular pentagon it is impossible to lay down with no gaps
I agree with Frank Seipp
For a tiling to be possible, the internal angle of the regular polygon must be a factor of 3 6 0 (since there are 3 6 0 ∘ in a circle).
Equilateral triangle: internal angle = 3 ( 3 − 2 ) × 1 8 0 = 6 0 ∘ and 3 6 0 ÷ 6 0 = 6
Square: internal angle = 4 ( 4 − 2 ) × 1 8 0 = 9 0 ∘ and 3 6 0 ÷ 9 0 = 4
Regular pentagon: internal angle = 5 ( 5 − 2 ) × 1 8 0 = 1 0 8 ∘ and 3 6 0 ÷ 1 0 8 = 3 3 1
Regular hexagon: internal angle = 6 ( 6 − 2 ) × 1 8 0 = 1 2 0 ∘ and 3 6 0 ÷ 1 2 0 = 3
Hence the regular pentagon is the only one of the four shapes which cannot be laid down with no gaps.