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id	title	challengeType	forumTopicId	dashedName
5900f49f1000cf542c50ffb1	Problem 306: Paper-strip Game	5	301960	problem-306-paper-strip-game

--description--

The following game is a classic example of Combinatorial Game Theory:

Two players start with a strip of n white squares and they take alternate turns. On each turn, a player picks two contiguous white squares and paints them black. The first player who cannot make a move loses.

n = 1: No valid moves, so the first player loses automatically.
n = 2: Only one valid move, after which the second player loses.
n = 3: Two valid moves, but both leave a situation where the second player loses.
n = 4: There are three valid moves for the first player; who is able to win the game by painting the two middle squares.
n = 5: Four valid moves for the first player (shown below in red); but no matter what the player does, the second player (blue) wins.

valid starting moves for strip with 5 squares

So, for 1 ≤ n ≤ 5, there are 3 values of n for which the first player can force a win.

Similarly, for 1 ≤ n ≤ 50, there are 40 values of n for which the first player can force a win.

For 1 ≤ n ≤ 1\\,000\\,000, how many values of n are there for which the first player can force a win?

--hints--

paperStripGame() should return 852938.

assert.strictEqual(paperStripGame(), 852938);

--seed--

--seed-contents--

function paperStripGame() {

  return true;
}

paperStripGame();

--solutions--

// solution required

1.7 KiB Raw Blame History