Enjoyable Enigmas #2: Who Wins the Matchstick Game?

[Enigman]

Can't find the old thread but I did find a new book of puzzles.

Two players, A and B, take turns in the following game. There is a pile of six matchsticks. At a turn, a player must take one or two sticks from the remaining pile. The player who takes the last stick wins. Player A makes the first move and each player always makes the best
possible move.
Who wins this game?

 

[DaveC426913]

Spoiler: My answer

B can always win. Let's see if I've got my logic right.

              6                          // start w 6
A       5         4                      // A takes 1 leaving 5, or A takes 2 leaving 4
B       3         3                      // B always takes enough to leave exactly 3
A     2   1     2   1                    // A takes 1 or 2, but it always leave 2 or 1
B     0   0     0   0

On the first move, if A takes 1, B should take 2.
If A takes 2, B should take 1.
Either way, here's 3 left.
There's nothing A can do but to leave either 1 or 2, meaning B can always win.

Thanks! I enjoyed that!

 

[mfb]

Or, more general:

Spoiler

If you can, always leave a multiple as 3 as remainder. No matter what the opponent does, you can repeat this, including zero where you win. All multiples of three are losing positions (you lose if it is your turn, you have to move to a winning position for the opponent), all others are winning positions (you win if it is your turn, you can go to a losing position).

 

[Enigman]

Both correct!

Next one:
In a boxing tournament there are 150 participants. First set consists of 75 matches, the second set of 27 matches with one player being given a bye and so on.
How many matches are held? What if the number of participants was N, where N is an arbitrary whole number?

 

[mfb]

Spoiler

Every match kicks one participant out of the tournament, one wins => N-1 matches

For N participants, can you make sure no one gets more than one bye?

 

[Enigman]

Spoiler

Match the participants who have gotten a bye with each other after a second person gets the bye.

If we have only a 4 gallon and a 6 gallon jug and a lake full of water, is it possible to get 1 gallon of water? If so, how?

 

[mfb]

Depends on the things we allow.

Spoiler

Just with filling them completely back and forth it is not possible, as all quantities are always a multiple of 2 gallons then. If we find some way to fill the 6 gallon jug to 50% (e.g. by tilting it, if it has a rectangular shape) it is possible (and easy).

 

[Enigman]

Correct.

Note: I haven't solved this one yet.
(Second Note: Probably why I am pushing the rock off the cliff in first place.)