Solve This IMPOSSIBLE Geometry Puzzle (It's Actually Easy!)

#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.

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[LOGICALLYYOURS]

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 :)

[MrMartinae]

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.

[joelwinter]

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).

[annekekramer]

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.

[emem]

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.

[SnowFaceChamcham]

Wow, finally a solution that just makes intuitive sense and doesn't make me angry. Bravo 😂

[primus.interpares]

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.

[makerpat]

Woodworkers will immediately note that this only works if the diameter of the hole doesn't cross over one of the cut lines.

[dhruvparate]

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.

[smalin]

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.

[Sailor]

What happens, if the holle overlaps the center?

[BryndanMeyerholtTheRealDeal]

Once you cut the wood, you'd need to use wood glue to keep them together.

[dl]

Close but not quite. What about the kerf?

[qcokay]

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:

22o2222




Since we can't "move the hole" one row down, we instead cut off a row at the bottom and put it on top:


22o2222



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:

1
2 22o222
3
4
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.

[qcokay]

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.

[simmavishnuramakrishnan]

Locate the cuts offset from the centre and cut the wood based on that and rearrange. Like shifting coordinates in a cartesean plane

[LD-dtsk]

Why would I be helping a drunk carpenter

[Peter_Jenner]

That doesn't work in the real world because of the kerf left by the thickness of the saw blade.

[cool-aquarian]

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.

[mikefochtman]

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.