Solve the Puzzle: Cover Bedroom with Two Carpets

[XOS123]

Help me solve this!

Hi,

Firstly, merry Christmas! Secondly, this has been bugging me for a few days now, wondering if any of you can help? Or if a similar teaser exists?


Right,

There's a bedroom measuring 12m x 9m (108 m2)

Two pieces of carpet:

1st one 10m x 10m (100 m2)
2nd one 8m x 1m (8 m2)

How can you cover the room exactly with the two pieces of carpet. You are only allowed to cut one of the pieces of carpet once.

It's easier to draw the room and the carpet on a piece of paper. I'm sure this can't be done(with only one cut anyways!)?

Anyone able to figure it out?

 

[Kittel Knight]

Are you allowed to divide one carpet in more than two parts ? (I'm considering only one cut)

 

[XOS123]

as in folding the carpet over and then cutting it?

nope :(

 

[davee123]

Well, for starters, the 10x10 carpet is obviously the one that needs to be cut, because it's the one that's too large for the room.

The 10x10 MUST be cut such that the "cut" section includes but is not limited to a 1x10 section of carpet. Therefore, the cut section's orientation is known-- its 10m edge must be parallel to the 12m side of the room. Therefore, we can also conclude that no portion of the cut section is more than 9m wide, and cannot include any part of the opposing 1x10 section. Thus, the "remaining" section's orientation is also known, because it must similarly contain a 1x10 section of carpet which must be parallel to the 12m side of the room.

As a result, there is a middle 8x10 section where the cut can be made. It clearly must start and end on opposite sides of the carpet, although I'm assuming that the cut may include some curves and/or angles (otherwise I think I can prove that it's impossible).

I believe that it's also true that the known 1x10 sections must be aligned against the 12m walls. If not, the space between the 1x10 section and the wall would have to be taken up by either the *other* 1x10 section, or both the 1x10 section and the 1x8 carpet. It can't be taken completely by the other 1x10 section, because in order to do so, it would have to cover a distance of 10m, which could only be achieved by completely severing the opposing 1x10 section from the middle 8x10 section, thus requiring 2 cuts. If both, it's a little difficult to describe, but it would appear that it's simply not possible-- certain areas would be required to be filled by one 1x10 section of the other which conflict.

Hence, I believe the 1x10 sections MUST be flush against the opposing 12m walls. Also, because the 1x8 section is insufficient to cover an entire edge of the 9m wall, the 1x10 sections MUST be placed in opposite corners, giving each of them opportunity to fully cover the 9m wall.

Unfortunately, this path of logic stops here. By process of deduction, it would appear that this arrangement requires certain areas of the carpet to belong to the respective 1x10 sections which creates an impossible situation. The 1x8 section of carpet is only allowed one of effectively two positions, both of which create the paradox.

====

Conclusion: As stated, it would appear impossible to me. I'm not 100% sure of that, but it sure seems that way. If, however, one were permitted to cut EACH carpet only once, it's certainly possible. Similarly, if folds are possible (effectively meaning multiple cuts) there are most likely possibilities.

DaveE

 

[belliott4488]

DaveE - I'm reaching the same conclusion, but only from repeated experimentation. I'm afraid I don't follow your argument.

Suppose the 1x8 piece were replaced by a 2x4 piece - I expect you'll agree that the puzzle could then be solved. I'm having trouble following your argument to see where it would break down for that case. (The "Unfortunately ..." paragraph is where you lost me.)

 

[davee123]

Suppose the 1x8 piece were replaced by a 2x4 piece - I expect you'll agree that the puzzle could then be solved.

Yep-- if it were 2x4 or if each carpet could be cut once.

I'm having trouble following your argument to see where it would break down for that case. (The "Unfortunately ..." paragraph is where you lost me.)

It's a little tough to describe. I'll try ASCII art:

10 x 10 carpet which has necessary 1x10 strips A & B which must be on separate sides
of the cut:
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB
AxxxxxxxxB

Placement in the room:
A--------  <-- row 1
A--------  <-- row 2
A-------B  <-- row 3
A-------B  <-- row 4
A-------B  <-- row 5
A-------B  <-- row 6
A-------B  <-- row 7
A-------B  <-- row 8
A-------B  <-- row 9
A-------B  <-- row 10
--------B  <-- row 11
--------B  <-- row 12

Now, assuming that the above is necessary (or the mirror version of it, see prior
explanation) the 1x8 carpet can go on row 1, 2, 11, or 12.  It cannot be placed in the
middle vertically, because this would require an imbalance in the A & B sections of the
10x10 carpet.  And effectively, this means it can only go on rows 1 or 2, since 11 & 12
are just mirror images of *that*.

This means that the entirety of rows 11 and 12 (the side opposite of where the 1x8
carpet can be placed) MUST be taken up by carpet piece B, hence resulting in:

A--------  <-- row 1
A--------  <-- row 2
A-------B  <-- row 3
A-------B  <-- row 4
A-------B  <-- row 5
A-------B  <-- row 6
A-------B  <-- row 7
A-------B  <-- row 8
A-------B  <-- row 9
A-------B  <-- row 10
BBBBBBBBB  <-- row 11
BBBBBBBBB  <-- row 12

However, this means that in re-assembling the 10x10 carpet, the last 2 rows of A *must*
have been totally stripped (except for the known 1x10 strip).  Hence, rows 9 and 10 must
be populated totally by B's:

A--------  <-- row 1
A--------  <-- row 2
A-------B  <-- row 3
A-------B  <-- row 4
A-------B  <-- row 5
A-------B  <-- row 6
A-------B  <-- row 7
A-------B  <-- row 8
ABBBBBBBB  <-- row 9
ABBBBBBBB  <-- row 10
BBBBBBBBB  <-- row 11
BBBBBBBBB  <-- row 12

Following this logic further, we get:

AAAAA----  <-- row 1
AAAAA----  <-- row 2
AAAABBBBB  <-- row 3
AAAABBBBB  <-- row 4
AAABBBBBB  <-- row 5
AAABBBBBB  <-- row 6
AABBBBBBB  <-- row 7
AABBBBBBB  <-- row 8
ABBBBBBBB  <-- row 9
ABBBBBBBB  <-- row 10
BBBBBBBBB  <-- row 11
BBBBBBBBB  <-- row 12

And the only plausible section left to fill is the 2x4 section at the upper right.

Anyway, that's the deduction that I was using. So, assuming that the A and B strips must be run flush with the wall on opposite corners (knowing nothing of the remaining 1x8 carpet other than it must be placed only in one place), we can prove that the remaining carpet section MUST be arrangable as a 2x4 in order to complete the problem.

But I'm not 100% sure of my other logic. I'm about 99% sure. I have this odd distaste about my proof that the 1x10 strips must be flush against opposing corners. Anyway, my money's on that this problem is impossible as stated.

Hm. There's also the possibility that I didn't consider of flipping over one of the cut sections so that it's its own mirror image. But I doubt that would help, somehow...

DaveE

 

[doodle]

Placement in the room:
A--------  <-- row 1
A--------  <-- row 2
A-------B  <-- row 3
A-------B  <-- row 4
A-------B  <-- row 5
A-------B  <-- row 6
A-------B  <-- row 7
A-------B  <-- row 8
A-------B  <-- row 9
A-------B  <-- row 10
--------B  <-- row 11
--------B  <-- row 12

Now, assuming that the above is necessary (or the mirror version of it, see prior
explanation) the 1x8 carpet can go on row 1, 2, 11, or 12.  It cannot be placed in the
middle vertically, because this would require an imbalance in the A & B sections of the
10x10 carpet.

Solution hidden below. As a hint, the 8x1 is placed exactly vertically at the center of the room. :)


1. Cut the 10x10 into 2 portions.
AAAAAAAAAB
AAAAAAAAAB
AAAABAAABB
AAAABAAABB
AAABBAABBB
AAABBAABBB
AABBBABBBB
AABBBABBBB
ABBBBBBBBB
ABBBBBBBBB

2. For portion A, move the whole piece 2m up and 1m to the right.
AAAAAAAAA
AAAAAAAAA
AAAA-AAAB
AAAA-AAAB
AAAB-AABB
AAAB-AABB
AABB-ABBB
AABB-ABBB
ABBB-BBBB
ABBB-BBBB
BBBBBBBBB
BBBBBBBBB

3. Insert the 8x1 at where the dashes are.

 

[frogman]

carpet puzzle

Here is a puzzle with the same dimensions, different set up.
Maybe your puzzle had been discombobulated in its telling before it reached you.


"[URL

 

[belliott4488]

Solution hidden below. As a hint, the 8x1 is placed exactly vertically at the center of the room. :)


1. Cut the 10x10 into 2 portions.
AAAAAAAAAB
AAAAAAAAAB
AAAABAAABB
AAAABAAABB
AAABBAABBB
AAABBAABBB
AABBBABBBB
AABBBABBBB
ABBBBBBBBB
ABBBBBBBBB

2. For portion A, move the whole piece 2m up and 1m to the right.
AAAAAAAAA
AAAAAAAAA
AAAA-AAAB
AAAA-AAAB
AAAB-AABB
AAAB-AABB
AABB-ABBB
AABB-ABBB
ABBB-BBBB
ABBB-BBBB
BBBBBBBBB
BBBBBBBBB

3. Insert the 8x1 at where the dashes are.

Very nice! I was wondering if a solution existed that might look something like that, but I couldn't come up with it. Did you come up with that yourself?

Happy New Year, BTW!

 

[doodle]

Happy New Year to you too; yes, after much experimentation with 'staircase' cuts and with the initial assumption that the 8x1 is located at the center of the room.

 

[Oerg]

wow lol, i was trying hard to solve it and i did not consider the 8times1 piece to be in the centre of the room at all, silly me haha.

 

[davee123]

Ahhh, I thought there was something still unsettling about my reasoning, I just couldn't see it... Damn, I've been too spoiled by people posting incomplete problems to this forum!

DaveE