Friday, March 21, 2014

Soft Landing

A problem for physics-major types.


At the instant of time shown, a particle has the velocity vector shown. Find the constant acceleration vector that brings the particle to the given landing point with the smallest possible speed of impact.

Introduce any variables you like.

(Note, this problem has nothing to do with gravity or any other forces. Alternatively, if you prefer to think of it this way instead, you might say that there was initially no gravity or any other force, and your task is to create a uniform force field of your own.)

A follow-up:

Given any initial position vector, and given any initial velocity vector, does there always exist a constant acceleration vector such that the particle passes through the origin?

Monday, March 17, 2014

Polishing off that pizza



I know that I said I wouldn't be posting the solution to Sicilian Solution III: Pepperoni Protocol, but I think there have been enough developments to warrant one more post....

Reader jeff worked through a few approaches, including finding the solution shown below. The picture of the solution comes to us from Harold Jacobs---it works because the cut goes through the center of the rectangle; goes through the center of one of the circles; and separates the other two circles.



By the way, I should say that Harold was also a colleague of Martin Gardner's, and he reminded me about the nice problems in Gardner's book Penrose Tiles to Trapdoor Ciphers. For example, how do you bisect both halves of a yin-yang symbol with a single cut? This problem dates back to at least 1917.

The "Penrose" of Gardner's book title is of course Sir Roger Penrose. It was both a privilege and a pleasure to be one of Roger's research students during the 1990s. In addition to his researches in physics and mathematics, Roger is widely read in the history of mathematics and knows a lot about mathematical games. He has held the W.W. Rouse Ball Professorship in Mathematics at Oxford, which is named for another mathematician with a historical sensibility and a taste for puzzles.

This morning, my wife found herself facing two hungry little girls, both of them eyeing the last chocolate-chip pancake. She kept the peace with this Solomonic cut:



Google "pancake theorem" for some results along these lines. Some other references you might want to check out:

Hope you enjoyed these forays into bisection! Lately I've been thinking about a physics puzzle...if I get anywhere with it, I'll post. :-)

Thursday, March 13, 2014

Sicilian Solution III: Pepperoni Protocol


This time it's artisanal.




Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of cheese to pepperoni?

(I won't be posting the answer to this one - feel free to post in the comments.)

This puzzle and the last one are part of an interesting genre called bisection problems. I looked around online, but I couldn't find a good collection of problems to link to, just various items scattered here and there. But some classic problems in this area include bisecting a triangle; bisecting the area and perimeter of a shape simultaneously; bisecting two shapes simultaneously, and so on. By googling around, I found this book which looks interesting. If others have any gems in this area, feel free to link in the comments.

Hope you enjoyed the Sicilian Trilogy! Thanks as always for all the comments and emails.

Tuesday, March 11, 2014

Sicilian Solution II: The Sicilian Solution Solution




Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of crust to cheese?

That was the puzzle in my last post. Reader jeff posted the correct solution in the comments and also sent the following diagram:



Another friend used the same method too.

It was an honor to receive an email about this puzzle from Harold Jacobs, author of Mathematics: A Human Endeavor. Harold mentioned several different ways to solve the problem, including an elegant approach based on the fact that a line bisects the area of a rectangle if and only if the line passes through the center of the rectangle. I reproduce his diagram here:


Both diagrams are reminiscent of the genre of "proofs without words" (a book of which is here).

What if the dimensions of the slice are made general?


We can apply Harold's method: To share the cheese equally, cut through the center of the cheese rectangle. To share the crust equally, share the whole equally, thus, cut through the center of the whole. These two centers determine the cut:


The equation of this line is easy to find. The center of the cheese is point (a + (b/2), a + (c/2)). The center of the whole is point ((a + b)/2, (a + b)/2). The line through these two points is immediately found using the two-point formula as y = x + (1/2)(c - b). This is the desired cut.

Just as in the specific case that we started with, the desired cut has a slope of 1.

So next time you're in a pizzeria and there's trouble brewing, just make a 45-degree cut passing through the center of the cheese rectangle!

Another way to think about it is to go for a 45-degree cut that is equidistant from the two corners.

One can view the y-intercept (1/2)(c - b) as the height of the parallelogram that the bottom player deserves to have, in view of the "excess rectangle" of dimensions (c - b) x (a + b) that sits in the top player's territory atop the (a + b) x (a + b) square:


The parallelogram is half as tall and just as wide as the dashed rectangle, so it has half the area. Similarly, the crusty part of the parallelogram (a smaller parallelogram) has half the area of the crusty part of the dashed rectangle (a smaller rectangle). And likewise the cheesy part of the parallelogram has half the area of the cheesy part of the dashed rectangle.

For the sake of amusement, I also tried to come up with the most abstruse solution possible:



Indeed, Harold told me that he liked this puzzle precisely because there are so many ways to solve it. He gave it to some mathematician friends to see what they come up with--if we hear back, I'll let you know.

Thanks for all the responses! And get ready for Sicilian Solution III: Pepperoni Protocol, coming soon to a blog near you!

UPDATE 1/8/2015 - fixed a typo or error in the solution image, thanks to a reader who pointed out the mistake.

Tuesday, March 4, 2014

The Sicilian Solution



Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of crust to cheese?