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# witzar

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2. ## Alice and Bob's marathon

Just brute force.

6. ## Removing pawns - the game

This is incorrect. This is correct.
7. ## Removing pawns - the game

This is incorrect.
8. ## Removing pawns - the game

This is incorrect (for example m=2, n=1).
9. ## Removing pawns - the game

@DejMar Of course I know the game Nim. That's a classic. My game is quite similar, but still different. Nim has beautiful analysis of the winning strategy, and the strategy turns out to be very simple. But neither the analysis of Nim nor the strategy applies to my game.
10. ## Removing pawns - the game

Here is the simple game I've invented (if someone invented it before, then I'm not aware of it): A pawn is placed on every square of m*n chessboard. Two players take alternate turns removing pawns. On each turn, a player removes one or more pawns. All pawns removed in a single turn have to be taken from the same row or the same column. The player who cannot make a move loses (alternatively: the player who takes the last pawn wins). Let's call the player who begins the "first player" and the other one the "second player". Which player (the first or the second) has a winning strategy depending on the chessboard dimensions? PS The game can be played on more interesting boards. Just draw some number of crossing lines on a sheet of paper and place a pawn on each point of intersection of the lines. The pawns removed in a single turn should come from a single line. I've tried to play this game on the pentagram (the star: five lines, ten points/pawns) with my 9 years old daughter. It was fun. Determining which player wins was even bigger fun.
11. ## Random points on x-axis

Which part? If you don't understand the first statement, then just look at the definition of probability. Let me quote from Wikipedia: "Probability is the measure of the likeliness that an event will occur." The key word here is measure. Measure is a non-negative value you assign to each measurable set of your space (the assignment should have some special properties). No measure - no probability, as simple as that, just by the mere definition. So let me state it once again: 1) we need to agree upon which subsets of our space are measurable, 2) we need do assign them measure, and only then 3) we can ask about probabilities of different events. Before steps 1) and 2) probability is not even defined, so questions like 3) are meaningless. PS Your example problem with rain and plots could be easily resolved by choosing a finite number of sets that should me measurable and assigning them proper measures. Same procedure cannot be done to solve the original problem.
12. ## Random points on x-axis

Asking about probabilities makes no sense unless there is a probability distribution. Unfortunately there is no reasonable distribution on R+. But an obvious distribution exists on interval [0,x], where x is any positive real number. One thing we can do in such case is to try to solve the problem for [0,x] and then see if the solution has a limit when we go with x to infinity. With this approach it's obvious that the probability we look for equals 0.5 for every x, so obviously it's limit is also 0.5 when x goes to infinity.

15. ## Total number of equilateral triangles

But I cheeted:

18. ## More coin tosses

But this is exactly what my second program does. It simulates the flips.
19. ## More coin tosses

And here is the straightforward approach: It gives same results.
20. ## More coin tosses

Here is a Java code I just wrote to confirm my calculations:

25. ## A tossing game you sure win

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