Drorbn at 01:41, 13 December 2012

2012-12-13T01:41:10Z

← Older revision		Revision as of 21:41, 12 December 2012
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	{{12-240/Navigation}}		{{12-240/Navigation}}
	===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 have 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?

	[[Image:12-240-DeckOfCards.png\|center]]

	See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

	{{12-240:Dror/Students Divider}}		{{12-240:Dror/Students Divider}}

Zetalda at 00:18, 25 October 2012

2012-10-25T00:18:35Z

New page

{{12-240/Navigation}}
===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 have 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?

[[Image:12-240-DeckOfCards.png|center]]

See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video.

{{12-240:Dror/Students Divider}}

== Theorems ==
1. If G generates, |G| <math>\ge \!\,</math> n and G contains a basis, |G|=n then G is a basis

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

3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> 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 : <math>\exist \!\,</math> N <math>\in \!\,</math> '''N''', <math>\forall \!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> N

has a maximal element: an element m <math>\in \!\,</math> A, <math>\forall\!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> m ( m + 1 <math>\notin \!\,</math> A )

== class note ==

<gallery>
Image:12-240-Oct-15-Page-1.jpg |page1
Image:12-240-Oct-15-Page-2.jpg |page2
Image:12-240-Oct-15-Page-3.jpg |page3
</gallery>

12-240/Classnotes for Tuesday October 16 - Revision history

Drorbn at 01:41, 13 December 2012

Zetalda at 00:18, 25 October 2012