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This is a tall order, but since this site is for everything mathematical, here goes:

What are some nice games/puzzles/problems/problem-like activities that would be appropriate for a three-five year old?

I ask this because I'm not particularly interested in trying to accelerate "number literacy" or prepare for "school mathematics". One may assume the three year old can count to 20, say, but the games don't necessarily need to involve number. I would like some activities that would introduce the surprise and delight of the problematic.

An example:

An article by a Russian mathematician (I can't remember the reference...) pointed out that if you let a young child count three objects and then separate the objects, the child will think that the number of objects has increased. When prompted to actually count the separated objects, the child was surprised and amazed to see that the number was the same.

The goal is to bring a bit of the joy of mathematical thought to a very young child. Any ideas?

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    list of 'X'; wiki-ized2011-02-02
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    @Jon: only moderators can make a question post community wiki (anyone can make community wiki answers). For future reference, if you see/make a post that should be community wiki, you can flag for moderator attention.2011-02-02
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    A similar question on MO that is now closed but contains some excellent answers may be found at http://mathoverflow.net/questions/19957/how-do-you-approach-your-childs-math-education2011-02-08
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    The excellent article you are referring to is by Alexandre Zvonkine and may be found [here](http://www.de.ufpe.br/~toom/others_articles/engeduc/Zvon-eng.pdf). I HIGHLY recommend it!2011-02-08
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    Get a bunch of solid cubical blocks in various sizes (1 unit on a side, 2 units on a side, 3 units on a side, etc) and a balancing scale. Have the kid seek a combination of two blocks that'll balance a third. Be prepared to buy expansion packs of larger and larger blocks. :)2011-02-09
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    Have you ever read Feynman's stories about going for walks in the woods with his dad; there is a lot of good material in there.2011-03-01
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    Great question Jon!2011-05-11
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    To complement @Matt's suggestion (about Feynman's father methods to educate him). You may appreciate the ['patterns method'](http://thenearbypen.blogspot.fr/2009/11/feynmans-father.html).2012-12-01
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    @jericson: you had a typo in your (excellent!) link. It should be [here](http://www.de.ufpe.br/~toom/others-articles/engeduc/Zvon-eng.pdf)2012-12-01

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