k-man

It seems that I disagree as there should always be a way for the surgeon to win. Looks like you're right, BMAD. I haven't taken my 9-long worm example far enough to see that the adversary cannot continue to maintain that pattern indefinitely as long as surgeon does the right thing.

Nice puzzle and title, BMAD.

- Would you have change for half-cent? - Sure! How would you like it? Is all in hundred dollar bills ok?

Considering that there is a $1 coin, the maximum is limited only by the availability of the $1 coins. Without using $1 coins, I don't see how you can have more than 99.

another proof 1) draw 3 additional lines parallel to each cevian passing through the remaining vertices of 2 triangles. 2) show that 6 other shaded triangles are mirror images of the green triangle and therefore have the same area 3) show that every part of a shaded triangle that lies outside of the big triangle has an equal unshaded part inside the big triangle. Midpoints of big triangle's sides are points of symmetry.

Ignore my previous post. I got my numbers mixed up.

I agree with gavinksong

@bonanova: You're right that if the wind direction has both head/tail component and cross component relative to the trajectory then analyzing only one or the other will result in correct conclusion that wind adds to travel time. However, the only crucial fact that I was trying to show in my previous post was that there exist closed paths (i.e. straight line A->B and back) and wind directions (i.e. perpendicular to the path) for which ignoring the cross wind component will result in incorrect conclusion - same time with and without wind. In other words, in some cases, the entire time penalty is swept under the rug.

I think your simplified analysis still doesn't address the cross wind correctly. Let's say that the closed path is from point A to point B and back in a straight line. It will take longer with the constant cross wind than without any wind.

I think Rainman is questioning the correctness of the OP as it's asking to show something that's not true. Let's take m=8 and n=10. Then w=GCD(8,10)=2. But GCD(37,39)=37 and not 3.

Not sure why this hasn't been answered yet, but I'll try to wrap up some loose ends here.... Edit: formatting

But, the OP says that the words can be less that 4 characters. It also doesn't say that the same character cannot be repeated inside the word.

@bonanova, I guess I interpreted the OP as only the points are given and we choose the equator.

Isn't this the same as

i am unsure by what you mean in your second answer. I interpreted your second question as "What 2-d shapes can be formed at the site of a second cut"?

My previous post was meant to be a clarifying question rather than a statement. Sorry if I caused more confusion.

Just to clarify... Since there are 168 primes in the range of 1...1000 I have 168 dice at my disposal. Each die N has the number of sides equal to the Nth prime number. Each side of the die N has numbers between 1 and Nth prime inclusive on it, so that the die with 997 sides has numbers 1...997. All dice are fair.

Assuming you meant a rectangular solid...