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16 votes

Making a 2n-digit number divisible by 9

I believe that Bob can win if (and only if) n is
SQLnoob's user avatar
  • 8,570
11 votes

Making a 2n-digit number divisible by 9

I'll add on to SQLnoob's answer and say Bob can't win if n isn't So she chooses $$ m = \begin{cases} 7-k & k < 7 \\ 1 & k = 7, 8 \end{cases} $$ For example,
Tyler Seacrest's user avatar
9 votes

Moving a knight to a new square each turn: who wins?

Here is a solution without the 4x2 grids (although it is a generalization of that solution). Pair up the squares such that the two squares in each pair are separated by a knight's move. On each of Bob'...
tehtmi's user avatar
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8 votes
Accepted

Paper, pencil and a bunch of bars

This game is So in this case If we take "rational" to mean that when either player had a winning move they took it then Now So
Gareth McCaughan's user avatar
8 votes
Accepted

Double or Take game

Answer to the non-bonus part: Proof: Bonus part: From 12 Hence
Gareth McCaughan's user avatar
8 votes

Removing green marbles from the table

Let losses be on $F(x). F(1) = 3; F(2) = 5, F(x) = F(x-1) + F(x-2)$ Suppose there are k marbles, and that all previous marbles N < k this Fibonacci formula decides correctly. The person to reduce ...
Lacklub's user avatar
  • 1,122
8 votes
Accepted

Best strategy for stick-taking game

Pretty much all Nim games can be solved by starting at the end and working backwards. Let's first solve the basic Nim, with only one pile, and simple actions (take 1-4). Let's enumerate the endgame ...
Bass's user avatar
  • 77.5k
7 votes
Accepted

A pile of chips involving primes

Here's a little Python program to test it yourself: https://repl.it/repls/ScientificIdenticalPixels And here's C++ code written by user @im_so_meta_even_this_acronym https://ideone.com/SfMHqC
Parseltongue's user avatar
  • 2,377
7 votes
Accepted

Marbles on a Mancala Board

You should To prove, let us first do the following
hexomino's user avatar
  • 136k
7 votes
Accepted

Removing green marbles from the table

My answer offers no proof, only truth. This puzzle is related to the base Fibonacci representation of numbers. In base b, the nth digit represents $b^{n-1}$. In base Fibonacci, the nth digit ...
Mike Earnest's user avatar
  • 32.4k
6 votes

Single-pile Nim with Three Players

Even though the question says I'm Bob, I'd like to start by stating my intention to be the Quetzalcoatlus, thanks very much. I believe that the key to this game is that Example game: However, ...
El-Guest's user avatar
  • 32.3k
6 votes

Single-pile Nim with Three Players

Partial strategy up to 11 candies. A is the player whose turn is next, B is the next player, and C is last. Don't think I can do it for 40 without turning this into a novel. But I have to say that, ...
Jafe's user avatar
  • 77.2k
6 votes

The 15 Pebbles Game

As with pretty much all the nim variants, this one can be solved by starting from the end and working backwards. With the original total number of stones being an odd number (15, as given in the title)...
Bass's user avatar
  • 77.5k
5 votes

The 15 Pebbles Game

The strategy is to do a move that In particular, for $15$ pebbles, your first move would be The reason this works is more interesting than with other single-pile nim variants.
Jaap Scherphuis's user avatar
5 votes
Accepted

Another variation of the game of Nim

1. Proof that all games will end after a finite number of steps Proof by induction: Induction hypothesis H: (not known to be true yet) ...
Bass's user avatar
  • 77.5k
5 votes
Accepted

Single-pile Nim with Three Players

I will assume that Given several equally desirable options, the players will randomly choose one. Doing so, it is easy to work out what happens when there are $n$ candies; If $n=1$, then player 1 ...
Mike Earnest's user avatar
  • 32.4k
4 votes

Can you help me understanding the Stones game?

Here's a clearer statement of the rules. Each player has two options: Remove a stone on this turn. Remove a stone on the next turn. This is because not removing a stone causes one to be removed next ...
Mike Earnest's user avatar
  • 32.4k
4 votes

Single-pile Nim with Three Players

Using the same assumption as @Oray, Now
sedrick's user avatar
  • 2,006
4 votes

Single-pile Nim with Three Players

With this information: To be perfectly rational and impartial, only wanting to maximise their own chance of winning. and it is assumed that If there were 5 candles If there were 6 candles, If ...
Oray's user avatar
  • 30.3k
4 votes

Double or Take game

Let’s see: I assume that all numbers must be greater than or equal to zero, otherwise the possibilities become too painful. If that was the intention, I’ll look harder. 1 2 3 4 5 6 7 8 9 10 ...
El-Guest's user avatar
  • 32.3k
4 votes
Accepted

A pile of chips involving powers of 2

Code to find the pattern: https://ideone.com/O5S4Qu (The third number printed out on each line is the winning move if the player to move is in a winning position)
sunfishho's user avatar
  • 938
4 votes

A pile of chips involving powers of 2

We have a winning strategy for: The first move is:
StephenTG's user avatar
  • 3,615
4 votes
Accepted

The 50 game between two players, selecting numbers between 1 and 10 inclusive + variations

Question 1: The status of a game can be encoded as a pair A-B, where B is the sum including the last number said, and A is the sum excluding it. Question 2:
AxiomaticSystem's user avatar
3 votes

Marbles on a Mancala Board

User hexomino already figured out the puzzle, and managed to actually find the very complicated path that was exactly how I came up with the game. To recap: The game itself is a lot easier to play ...
Bass's user avatar
  • 77.5k
3 votes
Accepted

Game Night at the Binomial Elks Club

Any permutation of the rows and columns is treated as an "equivalent" board. Part 1: Matt chooses to go second. (Case A) If Ben takes 2 or 3 tokens, then Matt can turn it into a 2x2 board and win. ...
Sherwin Lott's user avatar
3 votes
Accepted

The Box of Tic-Tacs

Partial (assuming I understood the requirements correctly): My assumptions: You (captive) start. You and the kidnapper always return on the even turn the same number of tic-tacs and you decide the ...
Marius's user avatar
  • 18.1k
3 votes

Three-player Nim

If everyone is trying to maximize their points: If both of your opponents work together to minimize your chances to win the game itself:
Nautilus's user avatar
  • 6,544
3 votes
Accepted

A short nim game

If you have to take exactly 5: If you can take a number from 1 up to 5 (which is what I'm assuming) If you can take a number from 0 up to 5 For the alternate game (which actually isn't too much ...
Excited Raichu's user avatar
3 votes
Accepted

Who wins the game?

Reasoning:
Mariia Mykhailova's user avatar
2 votes

Removing green marbles from the table

As described above, the first player loses if $N$, the initial number of marbles, is a Fibonacci number, and wins for any other integer $N > 1$. Proof follows. First, a definition. The remainder ...
user3294068's user avatar
  • 7,498

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