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Topic: Willywutang and the off-center wheel (Read 5671 times) |
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BNC
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Willywutang and the off-center wheel
« on: Jan 29th, 2003, 11:00am » |
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Willywutang looked at the piece of wood before him. It is a 10” diameter round, flat piece he wanted to use as a “wheel” in his art lesson homework. Alas! While drilling the center hole (1” diameter) , he sneezed, and the hole is way off-center (although, luckily, still within the “wheel” – the center of the hole is 3” away from the center of the wheel).
And then, revelation! Willy cuts the wheel into two parts, glued them again, and got a perfectly centered hole in the wheel. How did he do it?
Note: assume Willy has in his possession a magical cutter that cuts with zero width, and magic glue that can glue pieces with zero distance between them.
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How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
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prince
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Re: Willywutang and the off-center wheel
« Reply #1 on: Jan 29th, 2003, 11:23am » |
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Is there some trick here that I'm missing? I followed my first thought, and it works! That never happens, at least not with most riddles.
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BNC
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Re: Willywutang and the off-center wheel
« Reply #2 on: Jan 29th, 2003, 11:26am » |
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No trick.
It is under "easy"...
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How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
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aero guy
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Re: Willywutang and the off-center wheel
« Reply #3 on: Jan 29th, 2003, 12:00pm » |
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So we are talking about a cutter that doesn't need to start at the edge then?
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redPEPPER
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Re: Willywutang and the off-center wheel
« Reply #4 on: Jan 29th, 2003, 12:20pm » |
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You're probably thinking about making a bigger center hole? But what if this center must remain 1" diameter?
There is a solution for that, and the cut starts at an edge. Much trickier than centering a hole though, so hopefully Willy doesn't sneeze this time.
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James Fingas
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Re: Willywutang and the off-center wheel
« Reply #5 on: Jan 29th, 2003, 12:21pm » |
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As far as I can tell, there are actually two solutions for this one! And both are rather clever...
<edit> There are actually an infinite number of solutions. An infinite subset of them allow you to start from the edge. </edit>
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| « Last Edit: Jan 29th, 2003, 12:24pm by James Fingas » |
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redPEPPER
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Re: Willywutang and the off-center wheel
« Reply #6 on: Jan 29th, 2003, 12:35pm » |
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Okay, I got them. My solution was unique, but I figured out the infinite number of other solutions. And aero guy probably didn't mean widening the hole as I assumed, but rather one of these solutions.
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towr
wu::riddles Moderator
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Re: Willywutang and the off-center wheel
« Reply #7 on: Jan 29th, 2003, 1:16pm » |
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I'd think, cut a hole out of the center, 1" in diameter, then plug the hole 3" from the center with the piece you cut out..
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towr
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Re: Willywutang and the off-center wheel
« Reply #8 on: Jan 29th, 2003, 1:20pm » |
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Or you can cut any symmetric shape that has the hole in it, and put it in mirrored in such a way the hole is placed in the center.. (And there's an infinite ways to do that..even from the edge)
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James Fingas
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Re: Willywutang and the off-center wheel
« Reply #9 on: Jan 30th, 2003, 11:04am » |
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towr,
There's one more generic class of solutions that I've found (in addition to the ones that you listed).
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aero_guy
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Re: Willywutang and the off-center wheel
« Reply #10 on: Jan 31st, 2003, 12:13pm » |
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ok, my first guess was about the center cut which is why I mentioned the edge thing and considered it solved. I then came up with the flipped cut out section answer, but I'll be darned if I can figure out the other class.
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James Fingas
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Re: Willywutang and the off-center wheel
« Reply #11 on: Jan 31st, 2003, 1:39pm » |
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Some of the solutions mentioned so far actually satisfy the criteria of both classes.
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SWF
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Re: Willywutang and the off-center wheel
« Reply #12 on: Jan 31st, 2003, 8:03pm » |
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I guess it depends on how the solutions are classified. I would classify them into four different categories. Examples from two of these classifications haven't been mentioned yet. Both admit an infinite number of possiblities, and one of them could even be broken down into more subcategories.
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redPEPPER
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Re: Willywutang and the off-center wheel
« Reply #13 on: Feb 1st, 2003, 3:56am » |
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Let's say the hole is in A and needs to be in B instead. Let's call C the point in the middle of segment AB.
We can have an axial symmetry. Cut any shape that includes the hole and that's symmetrical along an axis that goes through C and that's perpendicular to AB. We can do so by starting at the edge or not. This shape can then be flipped and glued back.
Or we can have a central symmetry. Cut any shape that includes the hole and that has C as a center of symmetry. We can also start at the edge or not. This shape can be rotated 180° and glued back.
We can cut a shape that has two symmetry axis (AB and the perpendicular through C) that could be either flipped or rotated. But it's only a subcase of the two other possibilities. Technically, there would be two ways to paste them back though, so maybe that counts as a new category.
I think that covers it all up. I don't see any other possibility that doesn't fall in these categories. Anyone?
Edit: Okay, this didn't really include towr's idea of simply cutting the hole in the center and plugging the wrong hole with it (or a variation of this if the holes overlap).
Now I think I got them all :p
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| « Last Edit: Feb 1st, 2003, 4:01am by redPEPPER » |
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SWF
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Re: Willywutang and the off-center wheel
« Reply #14 on: Feb 1st, 2003, 4:57pm » |
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redPEPPER, you are very close. After thinking some more, maybe that fourth category I was thinking of should be included in some of the others. It all depends on if you consider one of the pieces to have a hole in it if there really isn't a hole in it (Hint: the 'hole' lies on the perimeter). However, even without that distinction, your descriptions do not cover all possbilities.
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SWF
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Re: Willywutang and the off-center wheel
« Reply #15 on: Mar 5th, 2003, 5:08pm » |
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Nobody has responded in over a month, so I better comment before forgetting what I was thinking. A few cases have not been mentioned. Whether the cutout intersects the edge of the circle or not doesn't really matter as long the conditions of one or more of the cases below are met.
The first category is biaxial symmetry, as described by redPEPPER. This shape must be big enough to include the hole. The piece gets flipped over to move the hole to the center. Depending on how you categorize, a subclass of this that has not been mentioned is a cutout with no hole in it, that gets flipped over. This cutout is not symmetrical, unless you consider the orginal hole to be a part of it. The removal of a round plug from the center and filling the old hole with it, could be thought of as a limiting case of this as you shrink a connecting ligament between the hole and the center to zero width.
There is also the case of when the cutout can be rotated through a certain angle (instead of being flipped over) and put back with the hole in the center. This case also has a subcategory where the cutout does not have a hole in it, so its perimeter doesn't necessarily have symmetry, it can be rotated and put back in to form a hole at the center. The point not mentioned before is that rotation angle before this piece is put back in does not need to be 180 degrees. If the hole is small enough and sufficiently near the center of the circle, any angle of 360 degrees divided by an integer greater than 1 would work. For the hole size and location given by this problem, a rotation of 180, 120, or 90 degrees is possible depending on the shape of the cutout.
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Fifis
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—Poet, composer, pianist.—
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Re: Willywutang and the off-center wheel
« Reply #16 on: Mar 9th, 2013, 10:00am » |
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Nobody has responded over ten years, so here is the solution.
Using a circle of the same radius (10") and its centre coinciding with the centre of the drilled hole, just mark the arc, cut the moon crescent and glue it to the other side.
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