Mind-Bending Puzzle That Will Challenge Your Intellect
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- เผยแพร่เมื่อ 30 ธ.ค. 2023
- #brainteasers :
A carpenter had a rectangular wooden board.
He wanted to have a circular hole exactly at the center, but he was drunk, so he drilled the hole inaccurately, off the mark.
Now he needs your help to fix this mistake, by using a hacksaw.
So, the objective is to cut the board into pieces and reassemble them to form a rectangle of the same dimensions as the original, but with the circular hole in the exact center.
You should follow certain rules:
1 - You can only perform straight cuts.
2 - You are not proficient in using a handsaw, so, you can only start cutting from the outer edges of the board.
The challenge is to achieve this task with a minimum number of straight cuts.
Puzzle:
What is the minimum number of straight cuts required to successfully accomplish the task?
This puzzle is known as Umbrella stand puzzle (aka, dissection puzzle).
The video teaches how to think in the correct direction to find the solution to a complex problem.
You are welcome to share puzzle, math problems or any topics for upcoming videos.
Gmail : logicreloaded@gmail.com
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There has been a wonderful debate in the comments section over an "alternate approach". Let me explain this.
"Alternate approach ": (Originally suggested by @qc1okay , kudos to him for this excellent idea. )
Let 'L' be the length and 'H' be the height of the board. Now, from the center of the hole, measure 'H/2' distance towards the bottom. Mark a horizontal line at this distance. Make a full length cut horizontally. We now have two pieces.
Don't touch the lower rectangle. Just focus on the upper rectangle where the hole is present.
From the center of the hole, measure 'L/2' distance towards the right. Mark a vertical line at this distance. Make a vertical cut. This upper rectangle gets split into 2 pieces. Also, we already had a lower rectangle.
These 3 pieces can be rearranged to form the same dimension rectangle with the hole at the center.
Please note: Here I am considering the hole to the left of and above the center. If you change the initial location of the hole, please make the horizontal and vertical cuts accordingly.
Importantly, the debate here was that, in some of the comments there was a little inaccuracy (but their end result was perfect) During the second cut, if you include the lower rectangle as well, then we will get four pieces. So, there would be 3 cuts (not just 2). Because the cuts refer to the number of times we physically slice through the board, not the number of lines we draw.
After the first full length cut, we got two pieces, so we'd need two additional cuts on the independent pieces to make them 4 pieces before you reassemble.
The point here is that, after the first cut, we can ignore the piece that doesn't have the hole, and perform the second cut only on the piece that has the hole. We still get the same result at the end after rearranging the 3 pieces.
I want to thank people below who took this alternate approach. Glad to see your comments guys.
@qc1okay
@MrMartinae06
@dhruvparate2696
@geeta172
@wholesomeparas1
Anybody who followed the alternate approach (the accurate one), you're absolutely on the right track :)
Yes, this is a correct summary of the simplest 2-cut solution. I want to add these points:
1) There is no need for a second cut at all if the offset hole happens to be on either of the L/2 or H/2 lines. The first cut finishes the job in my solution and in Ammar's flip-rectangle solution.
2) Ammar's flip-rectangle solution uses the least-cutting total distance.
3) Neither the issue in 2) nor the concepts of when to glue or whether a glued-on piece is still considered "separate" were part of the puzzle, so I didn't consider them. Now that I consider them, I think that since the puzzle clearly has an end result of a glued board being considered a single board for whatever purpose the drunk carpenter had for it, that clearly means that any time our cut crosses a glued-together area, that does not constitute an extra cut. My original solution from the first video meets all the criteria originally stated in the puzzle.
4) The puzzle didn't ask for the simplest solution but implied it was a real-world application, so I still prefer my simplest solution. Of course if additional criteria are added to the original wording, the best solution may be different. It's always tough to get good puzzles worded for a worldwide audience without giving away too much of the answer. As many mistakes as Ted-Ed and MindYourDecisions make in their TH-cam puzzles, we should admire how few occur here.
@@qc1okay I really appreciate that you settled the discussion with a good conclusion.
@@LOGICALLYYOURS Flip the board vertically and horizontally then the holes average will be in the middle!
..i can add something..
depends of what we consider a stright cut...if a strigt cut is one continues cut withour stopping no matter in what angle...we only need one cut...the blade makes a 90 degree turn (also assuming a infinite thin blade), if the blade has a width neither of the presented sulutions will make the squeare the same dimension...
so one (or possibly two cuts) that cut at C+(W/2) stright down to C+(H/2) changes direction and cuts to the opposite edge, ffor two cuts we just stops and do another cut starting at the opposite edge and C+(H/2), flip the pice and this two pices will fit together... thers no need to make it 3-pieces...
Edit: if we are allowed to make an arc cut we can make the piece the exact same dimmensins as befor the cuts with some nasty arc holes in it regadles of blade/tool width (3 cuts, or 4 if wer not allowed to stop)
Highlight from this comment: less than 2 may be enough
I also thought of the alternative solution. However, I didn't optimise it to be minimal. I thought of it as compacted with the torus topology. What that means isn't important here, except it gave me the solution below. My solution was to bring a part from one side to the opposite one and then do the same in the other direction. This would require 4 parts. The number of cuts depends on whether you can make a cut through several pieces at once. Moving the piece without the hole rather than cutting it is an optimisation. However, there's another optimisation that may be hard to notice. If you rotate the parts without the hole by 180°, they'll form one part. So, you don't have to finish the first cut. So, the number of cuts, once again, depends on how you count. I'd count that as 1.5 cuts
This was a fun problem. I solved it in 2 cuts by making the first cut all the way through then gluing the remaining piece on the other side, once it dried, making a perpendicular cut and gluing the remainder on the opposite side. I was assuming that cuts had to be completely through the board, bad paradigm as that wasn't a rule.
This was my solution and the solution of many other people after me. You might want to skim the comments before posting. And you don't need to glue pieces together until after you make both cuts. Your second cut does not go into the already-cut-off piece.
"I was assuming that cuts had to be completely through the board, bad paradigm as that wasn't a rule."
Why assume how cuts are to be made, instead of listening to the rules and watching the example of a valid cut? 0:56
That was my solution as well
Same solution here
As a woodworker this was an interesting problem--and the solution presented is academically/mathematically true--but in the real world any cut takes away some material (called the "kerf").
You can still re-center the hole in a board with two cuts in a different manner than shown here (I think it reflects the pinned solution in the comments here), but the size of the ending piece will necessarily be smaller (due to material removed by the kerf of the saw/blade).
Not to mention how the board is being joined back together. Better to present the problem as paper to be cut and taped together.
yeah, I was wondering if they would put in the caveat of the cut doesn't take away any thickness. Similar to physics problems that say "frictionless plane"
What an extremely complicated way to solve something so simple:
- cut off a part from the bottom, glue it to the tip
- cut off a part from the right side, glue it back to the left side
That's two straight cuts.
I wouldn't call the shown solution "extremely complicated". And yours isn't any simpler, you still have to take measurements to determine where to make your cuts.
Thats not simple ,in that way anybody can do this it should in same dimension after arranging whole at center !!
Wow, finally a solution that just makes intuitive sense and doesn't make me angry. Bravo 😂
Woodworkers will immediately note that this only works if the diameter of the hole doesn't cross over one of the cut lines.
Once you cut the wood, you'd need to use wood glue to keep them together.
I came with 2 cuts but different method: (hard to write easier to show)
1.Measure distance from left edge to center of a circle. Use the same distance from the circle to the right. Measure remaining strip at the right side, divide by half and cut. Move the strip and attach to the left side. And so you have the circle in the middle form left to right.
2. Than measure distance from upper edge , use the same distance from circle towards down. Than measure half of remaining bottom strip, cut it and move and attach to the top.
At first, it seemed like your solution was different from mine, but when you got to the end, I realized it was the same, except that mine was upside down from yours: in mine, the L-shaped piece rotated and the piece with the hole stayed in the same place.
That doesn't work in the real world because of the kerf left by the thickness of the saw blade.
You say, that a handsaw is the only tool you have to fix the problem. That means, you do not have anything to measure the offset of the hole from the center or mark, where to cut, so none of the suggested solutions will actually work. Pretty clear, that the minimum number of cuts is only two, but without the possibility of measuring of marking, you'll never fix it.
The back of the handsaw is almost always straight, so you could easily use the straight edge and your fingers to accurately mark distances.
@@HenrikMyrhaug Yeah, as accurately as your fingers still are, after using the handsaw, when you are drunk 😂
The cutting edge of the saw is also straight and it also has regularly spaced tines all the way along it to do the actual cutting. Just use that edge as the measuring device and count them. And if the measurement needed is longer than the saw blade, measure out one blade length and then use the saw to mark the edge you're measuring and carry on further. Simple.
@@ThePawsOfDeception Seems you have different handsaws than I have here in Germany. No markings, no measuring help...Just the teeth of the saw, which I could count, If course. Anyway, happy sawing 😉!
@@primus.interpares No, yours are the same as ours. When it comes to saws, I tend to use the terms 'tine' and 'tooth' interchangeably as they mean pretty much the same thing. Sorry for any confusion. So yes, just count the teeth.
Close but not quite. What about the kerf?
Before watching your answer, I came up with cutting fully across horizontally at the right point, then cutting vertically to form four pieces that could be rearranged. I wasn't thinking of that as three cuts, since I was thinking of like laying the board on a table saw and just running straight through, not able to 'stop' at a desired point. But yes, I agree with your pinned comment, that would technically be 3 cuts.
Great puzzle and a real 'head scratcher'. Thanks.
Thanks Mike, and I appreciate you also had a look at the pinned comment.
Awesome
I instead made two full cuts along the length & breath, on after another. Then put those pieces on opposite sides. Such that the centre is now equally away from all the edges.
Me too. Let the centre be O and it’s distance be x1,x2 from LHS and RHS of board and similarly y1,y2 be the distance of O from top edge and bottom edge.
Simply calculate (x1-x2)/2 if x1 is longer and attach. Repeat for y
I initially thought that it could also be an acceptable solution, but after careful analysis and verification I came up with this reasoning:
The cuts refer to the number of times you physically slice through the board, not the number of lines you draw. So, after the first full length cut, you get two pieces. Each of these pieces would require its own independent cut, making the total cut count as 3.
The rules are now very clear.. I appreciate you changed it. 👌🏻👍🏻
Thanks Niraj 😊
Love it!!!!!!!!!!!!!!!!!!!
What happens, if the holle overlaps the center?
I got same result, but with another way to calculate where to cut:
mark x and y on board such that the hole will be in center, and then cut at half of remaining length and width of board until they intersect and rotate cutout piece by 180 degrees and join them.
Perfect 👍 !!
Much better, well done sir 👍
Awesome, thank you!
Awesome Puzzle. Though one more solution could be to cut it completely from bottom side and rejoin it on top side to make the hole exactly between at the Vertical Center. Then do the same from right side and join the parts at left to make the hole exactly at the center
But in my solution even when number of cuts are two, "more length of the board is to be cut".
Very nice solution indeed. Another wonderful video. Keep it up bro
Amazing! The EXACT solution I posted two days ago. What a coincidence.
@@LOGICALLYYOURS True. Eagerly waiting for the more versions of this puzzle now :)
Yeah, had the same. Simplest task I've solved so far on this channel..
Yeah exactly what i did
Hi @wholesomeparas1,
Initially I thought the full length approach is valid, then after careful thinking i came up with this:
The cuts refer to the number of times you physically slice through the board, not the number of lines you draw. So, after the first full length cut, you get two pieces. Each of these pieces would require its own independent cut, making the total cut count as 3.
You did it very well, just that that extending the first cut to full length caused the board to split into two, after which you’d need to consider further two independent cuts.
That was the solution I considered too - but you have to allow for the width of the cut itself from the saw blade.
outstanding!!!
I correctly guessed two cuts but I would have needed to manipulate the physical pieces to figure out that I had to turn the piece around.
Why would I be helping a drunk carpenter
Here's a crystal-clear re-illustration of my edge-cuts solution that a few people may need:
1234567
8+abcde
FGHIJKL
mnopqrs
tuvwxyz
The hole is the plus-sign "+" occupying one unit in the thirty-five unit board. Cut #1 starts between m and t and goes to the right for seven positions. Now we have two pieces:
1234567
8+abcde
FGHIJKL
mnopqrs
tuvwxyz
The bottom edge-piece (seven units) is left alone, and cut #2 is to the four-by-seven main piece (twenty-eight units):
12345 67
8+abc de
FGHIJ KL
mnopq rs
The rightmost two columns are cut off from the rest, leaving three pieces total: a seven-unit bottom-edge piece (from cut #1), an eight-unit right-edge piece, and a twenty-unit main piece.
Now glue the right-edge piece to the left of the main piece:
67 12345
de 8+abc
KL FGHIJ
rs mnopq
And then glue the bottom-edge piece on top of the whole thing:
tuvwxyz
6712345
de8+abc
KLFGHIJ
rsmnopq
This 2-cut solution works 100% of the time: always cut off the bottom or top, depending on which is further away from the offset hole, and similarly then cut off the left or right edge. In fact, perhaps 20% of the time, only ONE cut is needed. If the hole is anywhere on the horizontal
"center line" or vertical "center-line", then you only cut once.
Mathematical problems presented as real world situations create unenvisaged consequences.
Cutting and refixing timber creates the consequence of having to reattach the pieces to form a whole sheet once more.
A second consequence which makes the given solution/s less than ideal is that timber has grain, and grain has direction.
The given solution leaves us with part of the grain running in the wrong direction. This has consequences in finishing, which will be required as there will be joins in the reassembled sheet.
To keep the grain running in one direction, first rip (technical term) a strip the length of the board to shift the hole in the vertical direction (as per your illustration), but before fixing the pieces make a cross cut (technical term) to adjust the hole horizontally. Transpose these two pieces and join, then transpose this with the ripped strip and join, making sure not to rotate it.
There will be a greater join length to repair but grain will run in the same direction.
Real world aside, I love the rotation solution.
I’d like To see what people can do with diagonal cuts.
so not only drink and drive
also remember donot drink and cut the board 😂😂😂
😁😁👌
Excellent puzzle
Thank you 🤗
Two cuts not going all the way through would not work because the width of the saw would not allow the piece to finish flush with the original edges.
100% makes perfect sense, as always.
Thanks 😊
@@LOGICALLYYOURS :)
This reworded version of this video from 2 days ago is much clearer. Here is the simpler solution I posted on that one, followed by Mr. Ammar's comment:
Yes, 2 cuts, but a more intuitive solution is simply 1) to cut enough off the bottom to attach to the top to fix the vertical displacement of the hole, and then 2) to cut enough off the right edge to attach to the left edge to fix the left-right displacement. Here's a 5 x 7 board with a hole at row2,col3 instead of at row3,col4:
1111111
22o2222
3333333
4444444
5555555
Since we can't "move the hole" one row down, we instead cut off a row at the bottom and put it on top:
5555555
1111111
22o2222
3333333
4444444
And since we can't "move the hole" one column over to the right, we instead cut off the rightmost column and put it on the left:
5555555
1 111111
2 22o222
3 333333
4 444444
So now the hole is at the (new) 3rd row and (new) 4th column. This is FAR easier than trying to figure out how to cut a rectangle and flip it around.
---------
LOGICALLY YOURS replied to qc1okay's comment
Mind-Bending Puzzle That Will Challenge Your Intellect
I appreciate your reply with the explanation. With full length cuts, the objective will be achieved certainly, but since its a puzzle, we can consider optimization (just like we considered optimizing the number of cuts, we should also consider minimizing the length of the cuts).
--------
Yes, the flipping-rectangle solution has the "least cutting length", but since that was never stated in the problem, I didn't address that issue. But with the goal of having a real-world solution that a carpenter could remember, mine is far superior.
Your solution covers the case when the "wrong" hole contains the center of the board - then the "flipping" solution fails. But how is a cut defined? It seems to me that you are making 3 cuts - the first horizontal one, and then 2 vertical cuts through the 2 wood pieces you now have?
@@plastonic Sorry, but I don't follow any of that. The flip-rectangle solution never fails so far as I can tell. And my illustration shows two cuts, not three. Look again. For cut #2, the "edge" is the seven 4s at the bottom of the 2nd picture, and the cut begins between its sixth and seventh 4. The cut goes up four positions, and the saw comes out above the row of 1s (which were the original top edge). Nothing has been glued together yet. Now we have three pieces to glue together: the row of seven 5s, the column of 1234, and the main piece. The row is glued at the top and the column is glued on the left.
I initially thought the approach you described with full length cuts could be valid. But after careful analysis and valuation I came up with below reasoning:
The cuts refer to the number of times you physically slice through the board, not the number of lines you draw. So, after the first full length cut, you get two pieces. Each of these pieces would require its own independent cut, making the total cut count as 3.
I really appreciate your approach, but at this point I beg to differ with that.
I would appreciate if you analyse my reasoning above and share your thoughts.
Hi @plastonic, true, the full length approach eventually produces 3 cuts in total.
Initially I thought the full length approach is valid, then after careful thinking i came up with this:
The cuts refer to the number of times you physically slice through the board, not the number of lines you draw. So, after the first full length cut, you get two pieces. Each of these pieces would require its own independent cut, making the total cut count as 3.
@@qc1okay The flip-rectangle solution fails, AFAICT, if the "wrong" hole is off the center just a little bit. Then your cutoff will contain circular angular bits.
The gist of my question was can you glue before you cut a second time?
waoh, nice problem
I've come up with the logic that something that isn't quite in the exact center can be considered as being in the center as a general area. For example, outer space can have a general center which we all are in. Thus we are all in the center of the universe.
Re-upload ??
Yes, the objective needed more clarity because several viewers didn’t get the problem statement correctly.
Question. I was considering a very similar solution but cutting the full width and full height and then reassembling the three pieces to form the rectangle with a hole in the center, BUT I realized that I had a hand saw but no measuring, marking, or straightedge devices to measure and mark perfectly straight lines. Much less trying to cut a perfectly straight line with a hand saw. But fortunately, the calibrated laser beams in my eyeglasses would be able to do the calibration and light-beam straight cuts required. Again BUT, I still wouldn't have any glue to fasten them together and the lasers can't help with that on wood. SORRY, we need more than just a handsaw. But still a fun challenge. Thank You.
great approach, my answer was 4, initially
2 right? you cut a bit of the bottom to move to the top, cut a bit from the right to move to the left, and then move the top part slightly to the left as well so it becomes rectangular.
1:26 Two cuts + some glue? To find the cuts, I would first find the exact center with two diagonals, then, make a horizontal line and a vertical line through the exact center; then find the two rectangles that does not have the hole on them, halve them again horisontally/vertically, and then cut these half way lines, then glue the sawn off pieces on the other side.
I brainfarted a bit; I would of course have to find half the distance between the hole and the farthest edge horizontally and vertically; not from the center.. But I think that would be equivalent, just cutting straight through into 4 pieces and reorganising rather than rotating.
Your explanation is great
Thanks a lot Ashutosh 😊
Yes, I can see how to do that with four cuts. Not sure, without watching, whether that's minimal. Cheers
[Edit] Never thought of that!
I misunderstood the parameters of this problem and was right but for the wrong reasons. Task failed successfully?
Since I was the first commenter to criticize the original wording of the instructions for this puzzle, I'll just say that this version fixes the problem nicely.
Highly appreciated !! 👍
👏👏👏👏
It takes 2 cuts.
Take half the width and cut that distance past the hole.
Do the same for the length.
You'll have an L shape and a rectangle
Rotate the rectangle 180˚and glue the two pieces back together.
Did you mean to mark half distance after the hole till the edge? Only then it would be a valid marking. If you take half the width of the board, you get a cut going through the Center.
@LOGICALLYYOURS yes, that's what I was trying to say. I wish youtube let us paste pictures, so I could just show my drawing.
This was a fun problem to work out.
But if the circle is already on one of the center lines its only 1 cut. So its only 2 cuts if the circle is not on one or both center lines
you've never mentioned that we can use a ruler, so i don't think your solution counts
You didn't say a measuring tape could be used.
I think this shows how familiarity with practical problems makes maths easier... I had this solved before the video started loading. It's increasingly significant with people starting to talk about how some examples in maths are harder for ethnic minorities to understand.
I dont know why people are talking about glue and kerf things in comments ite just puzzle how we can solve it with logic ,no need actually thinking about wood and all its just example only matter is its rectangle and whole which is to be placed at center without changing dimensions
But now you have a Frankenstein board and is useless. but FR it was a good thought experiment
why you are not uploading regular videos
Hey bro no new puzzles ?
I solved it by watching this vid to find the solution! 😆
Board will never be same dimensions as original as the saw removes a kerf (saw tooth) thickness when cutting.
Too bad cuts take away distance (i.e. blade thickness) and so the model is only so good. Math is fun!
This is the dumbest thing I've heard when thinking about a practical situation. A clean multi-directional cut with a handsaw?!
Superb problem, sadly I did it in 8 cuts 😅😂
Oh no! 😁
😀2 too, but not the same way H. N. Y. 🎉
I propose we start over
All in favor say aye
I solved this in like 3 seconds
2 cuts are enough.
This is simple but its not full solution. You should also prove that it is not possible to solve using single cut.
That is not provable since it's false! Some off-center holes are fixable with one cut:
1111
2++2
3333
4444
5555
To get that hole to the center, just cut off the bottom edge and glue it to the top:
5555
1111
2++2
3333
4444
@@qc1okayTrue! So the solution is wrong (or not complete).
too easy
I got the same solution as the video, ignoring the "kerf". The overall size of the board would be slightly smaller, but the hole would still be centered because the width of the cuts would cancel each other out after rearranging the cut pieces.
Too many assumptions. If the hole was taken out with a hole saw, just re-plug it with glue to preserve the shape. No new straight cuts necessary! If the carpenter was drunk anyway, how do we know the given hole is the correct size to begin with? Was it a pilot hole? Typically the hole in wood is for clearance purposes anyway. You could always make a larger hole from the center which would engulf the original hole on its tangent. Sure that hole could be quite big but again, no new straight cuts. Two solutions where 0 is the answer. Want one with 1 cut? A drunk carpenter who cannot drill in the middle of a board is no way going to solve this riddle while toasted. He will wait until morning and use a new piece. He's probably got a pile that he can make 1 straight cut to size and try again sober.
This is a picture of a board. Use Photoshop.