12-240/Classnotes for Thursday October 18

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Riddle Along

The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins.

Does this game has a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time?

12-240-DeckOfCards.png

See also a video and the transcript of that video.

Dror's notes above / Students' notes below

Theorems

1. If G generates, |G| \ge \!\, n and G contains a basis, |G|=n then G is a basis

2. If L is linearly independent, |L| \le \!\, n and L can be extended to be a basis. |L|=n => L is a basis.

3.W \subset \!\, V a subspace then W is finite dimensioned and dim W \le \!\, dim V

If dim W = dim V, then V = W If dim W < dim V, then any basis of W can be extended to be a basis of V

Proof of W is finite dimensioned:

Let L be a linearly independent subset of W which is of maximal size.

Fact about N

Every subset A of N, which is:

1. Non empty

2. Bounded : \exist \!\, N \in \!\, N, \forall \!\, a \in \!\, A, a \le \!\,

class note