Special announcement! Numberplay?s Gary Antonick will be on vacation for several weeks. The next Numberplay post will appear on Monday, Sept. 10.
Our puzzle this week comes from the math educator and Numberplay contributor Joshua Zucker, who uses it to illustrate a fundamental technique in problem solving: patience. ?Real problems require real patience,? says Mr. Zucker, who finds this instructive ?especially for kids, who may be used to solving everything in their math class in a few seconds.?
The puzzle is called Wolves and Sheep
On a 5?5 chessboard, place 5 wolves (who can move like chess queens) and 3 sheep so that all the sheep are safe from being eaten by the wolves.
That?s it.
Before getting to the recap of last week?s puzzle I?d like to introduce a remarkable new Numberplay tool.
Have you ever looked at a Numberplay puzzle with its batch of comments and wondered:
- How tough is this puzzle?
- Where are the real debates?
- What?s my own problem-solving style?
The answers to these questions can be found in Numberplay comments, but sifting through dozens of blog comments is like skimming a 20-protagonist mystery. You get the sense someone died, but you?re not quite sure who it was, exactly, or when it happened, or how the unfortunate incident occurred.
This is where Gary Hewitt?s new Numberplay Comment Fix comes in handy. This tool not only numbers comments and provides a handy filter-by-name feature, but also creates visual summaries of entire Numberplay conversations. These visual summaries provide surprising insight into the actual content of the discussion. The Comment Fix is your Pocket Poirot.
Let?s take last week?s Spaghetti Loops, for example, which involved knotting up an innocent plate of pasta.
CLICK IMAGE TO ENLARGE
Spaghetti Loops comment volume over time.
We haven?t yet looked at the solution yet, but before we do ? how tough was the problem? Could middle school students take it on? High school students?
Neither, according to the Comment Fix, which suggests Spaghetti Loops was brutal ? one of the more difficult puzzles seen in Numberplay. Maybe something for undergrads or even graduate students. Comment volume, which for an average puzzle peaks on the first day, started out low and climbed to its zenith midweek, suggesting some real grappling with whatever was thrown out as the initial answer.
CLICK IMAGE TO ENLARGE
3-Point Turn: Where?s the debate?
We?ll look a bit more at this grappling when we get to our Spaghetti Loops recap. Let?s first look at how the Comment Fix can identify another vital component of puzzle-solving: collaboration. Before reading on, check out the chart to the right. Can you find the debate?
The chart, which summarizes our recent 180-comment 3-Point Turn conversation, suggests a problem of moderate difficulty ? easier than Spaghetti Loops ? with an initial peak and subsequent rest. But there?s something else going on. Colors Green and Plum dominate the initial discussion and remain active long after everyone else has left the room. The two colors are clearly related, and their volume and duration would suggest some sort of collaboration. Something like a debate.
Several clicks to filter by name and it?s partially confirmed: Giovanni Ciriani and Dr W were pushing the puzzle past its original solution, with Dr W confirming graphically a result that Mr. Ciriani had initially reached by optimization. A little more reading shows that Mr. Ciriani was also continuing to experiment with something on his own: the geometry of parking in tight spaces.
This vehicular exploration is a fascinating read, by the way, and worth a small detour. What ignited a sustained curiosity in Dr W? What captivated Mr. Ciriani? I reached both by e-mail and received the following responses:
Dr W: I had at least four pieces of insight, all of them important, as follows:
1. It took a while for me to realize what Giovanni and Hans were doing by demanding that the steering be held constant to left or right extreme during the three-point maneuver. Frobozz?s remark also helped a lot.
2. Once I adopted the Giovanni-Hans constraint, I saw that each of the four wheels had to follow a concentric circular path, and these paths had identical radii in both extreme right and extreme left turnings. This led immediately to my three sets of four tangent concentric circles. It then became a game to manipulate the positions of these three circular sets to achieve minimum road width.
CLICK TO ENLARGE
Giovanni Ciriani
3-Point Turn according to Giovanni Ciriani.
Dr W
3-Point Turn according to Dr W.
3. The next bit of insight came with the realization that the angle of the lines connecting the centers of the turning circles (shown as the two violet lines in the diagram) was the only single variable I actually needed to alter to reach the minimum road configuration for any particular set of turning circle radii.
4. The final insight came when I realized the minimum road configuration was achieved when each of the four wheel paths had to touch an edge of minimal road just once.
5. A fifth insight came right afterward and was present in my thinking almost from the very start, even though I didn?t fully realize it then. The left edge of the road follows the straight line through the rear wheel stop (red) and tangent (green) points, and the right edge follows the line through the front wheel stop (black) and tangent (blue) points. I noticed that in configurations where the road width is not minimal, these lines are not parallel, or, alternatively, only two of these four points sufficed to define the road edges with the other two lying within.
Giovanni Ciriani: This was a good puzzle, because it applied to a real life situation. Thanks to the what-if challenges of a group of aficionados (frobozzz, Hans and Gary H. to name a few), the puzzle soon became more challenging than at first sight, and I decided to make it really apply to my own car.
CLICK TO ENLARGE
Giovanni Ciriani
Parallel Parking: The Theory.
Giovanni Ciriani
Parallel Parking: The Practice.
I measured the real geometry of the car, including positions of the axles, of the bumpers and the minimum turning radius. The optimization for a real car involved solving a system of four transcendental equations.
All in all this puzzle allowed me to improve my understanding of a 3-point U-turn, parallel parking, and parking in a narrow alley. I used to think that my parking skills were quite good, having honed them growing up in Italy where parking availability has always been tight. However, I was impressed to have improved my parking skills with Numberplay. Through the puzzle I gained a better understanding of car maneuvering. The photograph shows how close I got to the curb recently in one tight parallel-parking situation in just one straight back-up maneuver.
Recap: Spaghetti Loops
Last week we played around with Spaghetti Loops.
The puzzle: A plate of pasta contains 100 strands of spaghetti. Tie two loose ends. Keep doing this until there are no more loose ends. What is the expected number of loops at the end?
The solution: Hans said it perfectly:
So the answer is 3.3, or the sum of 1 + 1/3 + 1/5 + ? + 1/199. Elegant, beautiful, concise. Maybe a little too concise. What was going on? Here?s Baltimark?s explanation:
In ?my? layman?s terms. . .when you have N strands, you have a (1/(2N-1)) probability of adding one loop to the previous case by tying the new strand to itself (that?s my first term in the sum.). You have a (1 ? 1/(2N-1)) chance of being exactly in the previous case.
Expected value is, of course, the probability of X times X. So, back to my equations. . .
(1/(2N-1)) = probability of tying new strand to itself TIMES
(1 + E(N-1)) = new value if we tie new strand to itself PLUS
(1-1/(2N-1)) = probability of NOT tying new strand to itself TIMES
E(N-1) = old value, which is the case we are in if we don?t tie new strand to itself.
Marco Moriconi, who originally presented the problem, presents the solution similarly:
Call E(n) the expected number of loops in an n-strand plate. Pick one end of one of the strands and tie it to another loose end. With probability 1/(2n-1) you will tie it to the same strand, forming a 1-strand loop, and with probability (2n-2)/(2n-1) you tie it to some other strand.
In the first situation you are left with n-1 strands plus one loop, which will be neutral, that is, it will be a loop until the end of the procedure. Therefore, at the end we expect E(n-1) + 1 loops ? the n-1 strands are expected to form E(n-1) loops, and we have a 1-loop from the beginning.
In the second situation you have 2 strands attached to each other, forming a longer strand (size 2, if you wish). What we have now is a plate with n-1 strands, it doesn?t matter if one of the strands is longer than the others. In equations, we have:
E(n) = (1/(2n-1)) x (E(n-1) + 1) + ((2n-2)/(2n-1)) x E(n-1)
or
E(n) = E(n-1) + 1/(2n-1)This is a simple recurrence relation, whose solution is E(n) = 1 + 1/3 + 1/5 + ? + 1/(2n-1)
OK. Now this is starting to make sense. But if we did actually go about knotting 100 pieces of spaghetti we?d never get 3.3 loops, of course. We?d get some whole number. What would we tend to get? The most frequent outcome, as summarized by reader Gary, would be 3 loops, which would happen about 28% of the time.
After this main dish was devoured, frobozzz threw out a seemingly simple question:
Anyone given any thought to the distribution of lengths of the loops at the end of the process?
This kept the conversation rolling for much of the rest of the week.
Thank you to everyone who participated.
Notes
- I?d like to express my gratitude to Joshua Zucker for providing Wolves and Sheep. This puzzle was discussed recently at a Math Teachers? Circle co-led by Mr. Zucker at the American Institute of Mathematics in Palo Alto.
- I?d also like to thank Dr W and Giovanni Ciriani for sharing their experiences with the 3-Point Turn puzzle.
- Thank you as well to Marco Moriconi for recommending last week?s Pasta Loops. Dr. Moriconi picked up this puzzle from Peter Winkler?s Mathematical Mind-Benders.
- Activate numbered comments, Tex formulas, images and visual conversation summaries with Gary Hewitt?s Numberplay Comment Fix.
- Do you have any favorite puzzles or domestic animals? Please e-mail to Numberplay@NYTimes.com.
Source: http://feeds.nytimes.com/click.phdo?i=8cf57a6411c70f5f9fccebdeacdb6a3e
powerball numbers rory mcilroy spice girls david bowie taylor swift taylor swift
No comments:
Post a Comment
Note: Only a member of this blog may post a comment.