[Prof. Templeton]

There was once a very forgetful landscape contractor who was constantly “misrembering” the detailed instructions of his employers. He was always forgetting essential parts of what the customer had intended and instead would substitute his own without realizing it. How he stayed in business as long as he did is a mystery. Here is one example of his forgetfulness:

The customer had wanted a 10 meter by 10 meter square plot dug up and then fresh sod laid down.

“I want the lower right corner of the plot to start here where this tree used to be”, said the customer as he pointed to a stake pushed into the ground. “And I want the plot to be positioned so that the side over there more or less faces the mid morning sun”, he added, pointing toward the Southeast.

“No problem”, replied the contractor. “I’ll get to work as soon as I can get my equipment back here”.

The contractor returned to the site around 2PM and got to work. He dug up a 10 meter by 10 meter square centered on the stake in the ground and positioned one side facing the 2PM sun. Fortunately he had not laid any sod down yet before the customer came out to check the progress.

“What is this?” asked the customer. “This isn’t what I asked for at all”. He then proceeded to retell his instructions.

“Ohhhh, O.K., I got it now”, said the contractor. “I’ll just leave the area that I already dug up that coincides with what you want, dig up the rest, and put sod down on everthing.

“Just get it done”, said the customer and stormed off.

What is the maximum area that could have coincided between the two squares, or put another way, what is the least amount of extra sod that the contractor used?

[Guest]

So he dug out around the stake instead of having the stake in the bottom-right corner?

So he only dug 25% of what he was actually supposed to, and will have to waste 75 m^2 to fix the problem. Plus the 75m^2 that he wasted by doing it wrong...I'll say he wastes 150m^2 worth of sod. Or if he wants to make it a square then add 50 more m^2 for 200m^2(to fill in the bottom-left and top-right chunks.)

Edited June 4, 2009 by James8421

[Guest]

The area of overlap is always 25 meters sq.

[Guest]

I agree with KillerGuns.

Edited June 4, 2009 by hookemhorns

[Guest]

The area of overlap is always 25 meters sq.

Are you considering the angle of the square?

[Guest]

OK I take that back.

[Prof. Templeton]

Hmmm...maybe I should have included angles, although it doesn't look like anyone would have fallen for it.

[phaze]

Where in the world was he? Or are we going to assume that the world is flat?

[Guest]

I agree with KillerGuns on the landscaper's best possible outcome (the OP). But what's his worst case scenario? And where in the world and what time of year would that happen?

[Prof. Templeton]

I agree with KillerGuns on the landscaper's best possible outcome (the OP). But what's his worst case scenario? And where in the world and what time of year would that happen?

Indeed! What is the minimum area? It's pretty easy to visualize how we get the previous answers if the squares are oriented in the same direction. They overlap in the entire top left quadrant (top down view) of the original square. If the original square was off by 90 degrees, the new square would be rotated counter-clockwise and would overlap the entire lower right quadrant. What happens at 45 degrees then? or 22.5?

[Guest]

if he rotated it 45 degrees, would'nt that cut the 25m^2 of overlap in half?....12.5

[Guest]

if he rotated it 45 degrees, would'nt that cut the 25m^2 of overlap in half?....12.5

The overlapping area becomes a right triangle having a hypotenuse of 10m. Thus area is 50m².

[Prof. Templeton]

The overlapping area becomes a right triangle having a hypotenuse of 10m. Thus area is 50m².

you didn't mean to say 50m². The area of the triangle is 1/2 base x height.

[Guest]

you didn't mean to say 50m². The area of the triangle is 1/2 base x height.

When correcting James's answer, I myself made same error. Sure it's 25m2