Holes in the Table

Question

I was asked this question in an interview and though I was able to answer a part or few parts I'm not really sure if what I gave was the optimal answer. So here goes the puzzle:

A restaurant owner asked a Carpenter to make tables for him out of certain number of wooden blocks and boards. There are only a limited number of boards. He was asked to make a rectangular hole right in the middle of the table so that the owner would be able to embed the restaurant logo in it. The Carpenter was a drunk and while working on the last table (and last wooden board) he had to make, be made a blunder. He made the rectangular hole way off the center.

The restaurant owner fired him on the spot.

He then goes ahead and hires you, an inexperienced Carpenter, with a promise that if you could fix the blunder caused by the last Carpenter, you would get a huge contract for all the wood work needed for the rest of his establishments. This would be huge for you are completely broke right now.

1. How would you solve the problem?
2. How would you solve the problem if the hole made was made at a 45 degree angle and needs to be corrected as well?

3. How would you solve the problem if both the table and the hole were oval in shape?

Rules:

Since you are inexperienced and an amateur and you only have a hacksaw, you can only start cutting along the outside edges of the table.

For 1. You can only make two cuts as otherwise it would ruin the look of the table.

EDIT:

The table has to be the same size as the rest of the tables. The size of the table is specified by the owner.

Need to give the most optimal solution and if possible prove it's optimality. Optimal solution will have the minimal cumulative length of the cuts

Answer 1

For part 1:

Measure the table's edges, and determine the mid-points. Measure the x and y distance from the centre of the hole (contrary to the image - got that wrong) to the edges of the table. Measure the same distances from the centre point. Cut to this second point (i.e. along the black lines), and then rotate the section that you've cut out by 180 degrees.

For part 2:

This one's actually not too hard to do, but difficult to explain - in essence, you can cut material equal to the hole in size from the middle, and place it where the actual hole is. See the diagram (and assume the central shape is the same size as the hole, please!) - if done in the correct order, all cuts are cuts from the edge


A potential cutting order is now included (edit: only cuts 1,2,3 and 4 needed). Once done, place the rectangle bounded by 1,2,3,4 in the hole bounded by 6,7,8,9

For part 3:

Much as for part 1, this involves cutting a rectangle that includes the hole and its distance from the edge. Measure the x and y co-ordinates of the centre of the board, and the x and y distances from the centre of the hole to a point on the outer edge of the ellipse. Measure those same distances from the centre of the board.
Cut out (see black lines) the rectangle described by these two points and the x and y planes.
You will need to cut two extra lines to allow you to cut the whole shape free, as you can only cut from the outside.
Then rotate the rectangle by 180 degrees, replace it, and return the other two removed segments to complete the table

Answer 2

One cut solution for #3.

Cut and turn the green part for 180deg. (Orange - original hole position)

Answer 3

I thought @LogicianWithAHat has answered the question quite neatly.

I think Q2 could be solved like this (it is not to scale, but you get the idea):

Put the triangle on top of the triangle with hole and the trapezium to the left.

Answer 4

I think people covered this, but I'll give it a shot for part one and two.

Part 1:

Start a cut from the top of the table, to the left of center by the distance from the current hole to the right edge of the board, and cut until you're at the center of the board, plus the distance from the current hole top edge to the top edge of the board. Continue cutting all the way to the right. Take out the piece you cut, flip over the x and y-axes, and reinsert. Summary: Make one L-shaped cut, rotate the cut out rectangle by 180 degrees, and re-insert.

Part 2:

Make one circular cut starting at the top of the board above the current hole (so that the rectangular hole is encompassed within a circle, and that circle's radius extends to the edges of the current board. Next, make essentially the same cut as in Part 1 so that you can rotate a large section of the board, and re-insert the circular cut out in the center, rotating it by 45 degrees. Summary: Make a circular cut around the hole, and an L-shaped cut across the majority of the board. Rotate L-shaped cut by 180 degrees, circular cut by 45 degrees, and re-insert.

Sorry for the crappy quality, I can't photoshop.

I have no idea about Part 3.

EDIT: As a comment to the question (since I don't have the privilege yet)--I was interpreting the third part to mean that you are given an ovular table with an ovular hole and still have to accomplish the same objective of turning it into a rectangular table with a rectangular hole; yet the #1 upvoted answer right now (for the part 3 solution) seems to interpret as- just get the ovular hole into the center of the ovular table (which is much easier IMO). Can anyone clarify what the objective for part 3 is?

Answer 5

Assuming I can cut more than twice for 2 and 3

For 1

Make two cuts so the hole is in the middle of the smaller piece cut out of the original table, the table will be smaller, but seeing that the owner never specified how large the table had to be you can just cut it like that.

For 2

Do the same as in 1 but make more cuts to get the desired result.

For 3

Dot the same as in 1 and 2 but make it a round cut so it retains the shape.

For more information

You could use the remaining pieces of wood to craft equal tables in shape just different in size.

Answer 6

Even if I cannot prove, this looks pretty much optimal:

same for 2 and 3.

I'm not sure if that can be made with an hacksaw: it probably doesn't like the sharp bends. To avoid them we could go for something like this, which is obviously suboptimal, but still possibly better than other solutions (depends on the size and on the initial placement of the hole).

To start cutting around the hole I assume that it is possible to insert the hacksaw in the hole, otherwise it should look pretty much as the other cut.

Answer 7

_{(On mobile right now, so I can't validate my answer or draw pretty pictures)}

Assuming your amateur skills extend to joinery, you could, for the first:

cut a thick L shaped section off from the lower left, leaving a kind of smaller version of the table/hole, the rotate the L and reattach it to the other side (you'd just have to do the maths to make sure the amount you removed plus the amount on the far side of the hole equaled the amount that remained on the near side).

For the second (edited):

a similar approach could be used, but first you need to cut on a 45 degree diagonal (parallel to the hole), starting at the bottom right corner such that you come in the same distance as the short edge. Then turn 90 degrees to the right and finish the cut. This will produce a kite shape which can be flipped over, effectively rotating the hole by 45 (your two cuts were parallel to the hole, with the first cut becoming the new short edge). Now you've effectively got a table like in number one, with the hole the correct angle, but off center, so we just repeat those steps.

For the third:

Not 100% sure on this one, because it feels like you'd struggle to maintain the appropriate curve, but you could possibly repeat the same process, only this time initially cutting a crescent shape section from the bottom left.

Answer 8

3 is simple assuming an L cut is one cut.