BMAD

I think you will discover many such winning lines, but now consider the line (or summation of line lengths), and try to find it/them that is/are needed to create the shortest guranteed win

The trick that kids in my homeland soon discover is the rule of crossing. As long as the line runs tangent to another line (or itself) then it did not cross but did effectively cut off some of the board. This distinction may not exist in English, and I am sorry if it confused anyone. I once got in a heated discussion with someone over what the word cross means. As my friend meant touching is the same as crossing, but for my people, crossing requires the passage over (beyond) the line.

I am sorry, I am working on translating to English better. I mean to say that the numbers you use must contain the sequence from 1 to some N but don't have to start that way. So 1, 3, 5, 2, and 4 are fine since it contains numbers 1-5 and no first hand neighbors.

these are the solutions. Well done everyone!

i should have been more clear in the op, the sequence must start at 1 and go to some n greater than 1 (consecutively).

the sequence must start at 1 and go to some n (consecutively) where n does not equal 1.

elaborate please.

is a 5 sequence possible?

We have two circles. One circle has a diameter of 12 cm's while another is D cm's. The circle with the diameter of 12 cm's fits inside the other circle. If rolled within the larger circle, the smaller circle rotates three times before coming back to rest at the same starting point. That is, Point p in the smaller circle starts and stop at Point P' on the larger circle after three revolutions. What is the diameter of the bigger circle?

You are conducting a survey where the question is somewhat embarrassing: have you shoplifted within the past 12 months? You realize that it might be difficult to get honest answers, so you ask your friend the psychology student for advice. He tells you the following trick: Ask each person to flip a coin and tell them that if the coin land heads, they should answer the question with a lie, if the coin lands on tails, they should answer 'yes'. As the person agrees to this before the outcome of the flip is revealed, people would be more likely to participate in such questions. Should you follow your friend's advice? Can you get any meaningful statistic by applying this method?

yes i agree. sadly monte carlo only gives us our best odds, though our 'best odds' always seems to be, to not play in the first place

Create the shortest possible sequence such that no number is next to its first-hand or second-hand neighbors (e.g., the number 3 cannot be next to numbers 1, 2, 4, or 5, which are first- and second- neighbors).

Create the shortest possible sequence (using numbers from 1,2,...n) such that no number is next to its first-hand neighbors (e.g. the number 2 cannot be next to numbers 1 and 3, which are its first-hand neighbors). e.g. 1,6,3,5,2,4,7 Here is a sequence using 7 numbers that meets the requirement.

There is a simple game that I used to play as a kid in my home country. Here is a simplified version. There is a game board with 4 dots on three levels setup as follows: * * * * * * * * * * * * One player moves first (usually the youngest) and both follow the same rules. Connect one dot to any other dot. When connecting two dots you may not cross over another dot, you may not cross a line made from a prior connection, nor can you cross over your own line. Dots that are already connected are 'off limits' from reuse. Dots may be connected using any type of line (curve, straight, etc.) as long as it is continuous. The player that cannot make a move is the loser. Assuming that the youngest person goes first, which player has the winning strategy to guarantee victory?

How much does a player stand to lose per round they play?

I have three possible and perfectly acceptable solutions to this question each depending on what assumptions in your model you choose to make. Only two of the three possibilities have been correctly (at least according to my logic) found. Joyandwarmfuzzies- 1/3 K-Man - 1/2 But there is still another possibility that I would like examined.

Yes, it's a different result then. I took the "directly behind" interpretation. It also took a second to see "patter" meant "pattern" I'm thinking there is further richness (or simplicity) in the solution, my answer was in the manner of first thoughts. Yes, the OP intended the individuals to move directly behind the person they identified. The answer has been given by one of the participants here but i will hold off a little longer in siding with a particular answer to allow some additional thinking on the problem.

very good!

care to explain your logic?

Yes, but the particle is not moving at constant speed . It starts out with no momentum (perhaps I should have specified that, but I thought it was implied), so its momentum is dictated by gravity. [spoiler=nice hint ] Since the particle starts with a speed of 0, the angle at that point must be nothing, hence it is is tangent to a vertical line at the origin. There fore the speed must reach a max value when the trajectory becomes horizontal and the angle 90 degrees. This screams for a cycloid

An examination in three subjects, Algebra, Biology, and Chemistry was taken by 41 students. The following table shows how many students failed in each single subject and in their various combinations: Subject: A B C AB AC BC ABC Failed: 12 5 8 2 6 3 1 For instance, 5 students failed in Biology, among whom there were 3 failing both in Biology and in Chemistry, and just one of these 3 failed in all three subjects. How many students passed in all three subjects?

a ball point pen was priced at 50 cents in the store opposite of a school but found few buyers. When, however, the store reduced its price, the whole remaining stock was sold for $31.93. What was the reduced price?

I have a stamp collection. It consists of three books. Two tenths of my stamps are in the first book, several sevenths in the second book, and there are 303 stamps in the third book. How many stamps do I own?

Bob, Peter, and Paul desire to travel together. Peter and Paul dislike each other and as a result refuse to ride with one another. Bob has agreed to drive each one following a creative system. With the car covering c miles per hour, the three people adopted the following scheme: Paul rides in the car with Bob, Peter walks. After a while, Bob drops off Paul, who begins walking; Bob returns to pick up Peter, and then Bob and Peter ride in the car till they overtake Paul. At this point, they change: Paul rides and Peter walks just as they started and the whole procedure is repeated as often as necessary. Both Paul and Peter walk at the same speed, covering p miles per hour. A. How much progress (how many miles) does the company make per hour? B. What fraction of the travel time does the car carry only person?