Mathematics problems for very young children? 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?
 A: It is never too early to introduce count-by. Count by 1/2: 1/2, 1. Count by 1/4: 1/4, 1/2, 3/4, 1. Count by eggs: 1/12, 1/6, 1/4, 1/3, 5/12, 1/2, 7/12, 2/3, 3/4, 5/6, 11/12, one dozen. Count by 2s: 2,4,6,8,10, count by 3s: 3,6,9,12, etc. While pushing on the swing count by 1s to 10, 2s to 20, threes to 30, 4s to 40, 5s to 50, 6s to 60, 7s to 70, etc, until you get to 100. You do the counting, and teach the sequences. You want to get those hard wired like the zyxwvutsrqponmlkjihgfedcba sequence or its reverse. Always ask your child what is the derivative of a constant when you are putting him/her on your shoulders. 
If you have square tiles in the kitchen teach about area. Play with blocks and build. Tie knots in rope and especially do math as a play activity.
A: This is complicated.. I find it really hard to make my 4 years old niece understand the concept of quantity. She can count, but she still thinks the numbers are like names for the objects. I remove the "one" object and ask her to recount and she goes "two, three four"
So maybe the best way to introduce math elements is with visual things, like jigsaw puzzles for kids, with big wooden pieces. Matching shapes and colors.
A: One thing you should try is to do a web search for Montessori equipment, much of which is specifically for the purpose of teaching mathematical concepts to 3–6 year olds.  The equipment itself is expensive, but it will give you ideas. For example, see here.
One example that comes to mind is that there is a tray designed to hold ten wooden cylinders.  Each cylinder has the same height, but a different diameter, and a little knob on top for picking it up; this is the top tray in the picture below. The tray has ten sockets into which the cylinders fit exactly, so that only their tops are visible. You present the three-year-old with the tray with the ten knobs showing, and she will immediately start to take out the cylinders. (The knobs encourage a pincer grip, which is another important skill for three-year-olds.)  Then she will examine them, and try to put them back. Some won't go in at all.  Some will go in but will be obviously too small for the sockets. A cylinder in the right socket will fit perfectly. She can try moving the cylinders around until she gets them all right; it's obvious to her senses of sight and touch when they aren't right, and then when they are.  When she gets  all the cylinders back in the sockets, she can take them out and do it again.

When she has mastered the first tray of cylinders, there is a second tray, where each cylinder has the same diameter, but a different height; this is shown at the bottom of the picture.  This tray is a little harder, because a short cylinder will fit into a long hole, and the error won't be apparent until later, when the displaced long cylinder has nowhere to go.
Then there is a third tray, where the shortest cylinder has the narrowest diameter and the tallest one has the largest diameter, and a fourth tray where the shortest cylinder has the widest diameter and the longest cylinder  has the narrowest diameter.
There are other activities to do with the cylinders.   For example, take them out of the tray, mix them up, and then arrange them in order on the table.

Other typical Montessori mathematics equipment:

*

*A basket with ten blue geometric solids: cube, cylinder, cone, sphere, rectangular prism, triangular prism, square pyramid, triangular pyramid, ellipsoid, and ovoid; and correspondingly-shaped cards with figures outlined in blue.


*A set of measuring sticks in 10, 20, … 100 cm lengths, with the 10 cm segments painted in alternating red and blue stripes.


*A "brown staircase" of $n$×$n$×10 cm brown wooden blocks and a "pink tower" of $n$×$n$×$n$ pink wooden blocks, for $n$ from 1 to 10.


*"Binomial" and "trinomial" cubes:

Montessori primary guids explains how these materials are used.
A: I like games with a mathematical/logical style.  Rush Hour is a favorite around here (though anything beyond the very easiest few are quite challenging for a five-year-old), and there are lots of similar ones.  The same company also makes a peg-jumping game called Hoppers which is very appropriate for quite young kids - even setting up the pegs properly is a good exercise for a 3 or 4 year old.  
Just to prove I'm not a corporate shill, I'll mention that lots of easy card games promote classification and basic addition etc. - try playing Uno or Crazy Eights (the latter can be completely generic, of course). If your child is able to do very easy adding by counting objects, there is a game called Sleeping Queens which, though it says 8 and up, is totally doable by many (most?) six-year-olds, and even those who can't add can still often win just by being able to match (since there is a heavy dose of luck involved); don't underestimate how useful it is to not have to LET a child win in a game that isn't 100% deterministic (as opposed to Candyland post-shuffle...).
A: You can let him count things, like a number of apples or bananas. It will be easier if the objects are something the kid knows well, nothing abstract.
Also you can let him sort pens or something by their height, it will be interesting to see if he develops some technique to do this. 
If he is very talented (probably not as he is just 3 years old) you could go on by letting him add small numbers or for example ask him how he would give 6 pieces of cake to 3 persons so everyone would get the same amount. Also pattern recognition is an important more advanced skill in mathematics.
A: I think any type of sorting or matching would be appropriate for a 3 year old.  Dominoes, or cards that match geometric distribution of numbers are a great way to building connections (this teaches about equivalence).  Algebra tiles would also work (e.g. make as many squares as you can with one large square, a few rectangles and a few small squares.) Note: algebra tiles come in the aforementioned denominations.  If this is done intentionally, you could draw connections later in life to algebra.  
A: I heard somewhere about colouring graphs with a minimal number of colours. My son was four years old when we looked at this and he loved it! I am sure it is interesting for a three year old too.
A: Three years old seems very young to do more than instill a sense of wonder about things that happen in the world which it will turn out in the future are related to mathematics. If you trust a three year old with a scissors than try having the child cut down the middle a long thin rectangle after its thin edges are taped, once with a half-turn before taping and once without the half-turn before taping. There are lots of nice paper cutting and paper folding examples that are fun for kids. I have done this, including the Moebius band work, with kindergarten children: http://www.york.cuny.edu/~malk/kindergarten/kindergarten.html
A: Well, I have a three year old granddaughter, and she can (more or less) count up to 100.  She pretty much did it for me, with only a few glitches.  I asked her if 100 was the biggest number, and she thought the answer was "yes".  I just accepted that.  It'll be interesting to see when the answer changes to "no".
A: Young children usually recognize shapes and geometric objects better than they understand the abstract concept of number. One possibility I've always wanted to try are arranging blocks. You can arrange blocks into two-dimensional grids to make rectangles. You can try to give a child the problem of making such a rectangle by just arranging the blocks in a grid.
You could try showing such a child that certain numbers cannot be arranged into rectangles without one of the sides having length 1, which corresponds to prime factorization. You don't have to explain prime factorization or factorization at all, but the experience might stick and later when the child grows up he or she might find the memory useful for understanding factorization.
A: My suggestion is similar to that of oosterwal.
My kids liked to hear "stories" of the following kind: If there are three trucks in the yard, and Thomas brings two more, how many are there now?  If Diesel comes and takes away one, then how many are there?   My experience was that for three and four year olds, simple arithmetic with numbers less than five was in reach,
and that visualizing the objects in terms of story settings and characters that they liked made it interesting and fun for them.  (Even just adding or subtracting one object is non-trivial at that age, and I think that practicing it helps develop the sense of numbers as cardinalities, and not just names.)
A: I am sure you have likely come across most of these, but I'll post them just in case. Have fun!
http://teachingmybabytoread.blog.com/math/math-for-two-and-three-year-olds/
http://www.ehow.com/way_5157467_math-games-year-olds.html
http://www.delmarlearning.com/companions/content/1401872352/Resources/OLC_Math_Section1.pdf
http://www.mathatplay.org/resources_pwm.html
http://www.topmarks.co.uk/maths-games/3-5-years/counting
http://www.education.com/reference/article/Ref_Building_Early_Math/
http://preschooler.thebump.com/fun-ways-teach-math-three-year-old-2795.html
http://www.zerotothree.org/child-development/early-development/supporting-early-math-skills.html
http://books.google.com/books/about/Teaching_Mathematics_3_5.html?id=T3mBVX9W0_4C
http://online.wsj.com/article/SB10000872396390444914904577615690632669590.html
www.youtube.com/watch?v=YI-2qQm1d44
http://www.pbs.org/parents/childdevelopmenttracker/three/mathematics.html
-A
A: My limited experience with three-year-olds who can count to 20 suggests that the counting is simply a recitation. Can your particular three-year-old reliably count the number of matches (10-20) in a matchbox? That would be a rare talent, and one worth encouraging.
Edited to add: An article in today's New Scientist agrees with this assessment:

Children learn this count list well before they actually understand that "four" refers to four objects rather than three or six, says Michael Frank at Stanford University in California.

A: Try some simple graph theory. Show that them you can draw edges between vertices without drawing over the same edge twice, and explain the general result of Euler in a way they understand.
Edit:
Or just show them Bridges of königsberg, or what it's called, and challenge them and let them try to figure it out themselves. Then guide them etc.
A: I used to do simple mental addition and subtraction with my kids when they were that age.  Questions like:  There are five cookies in this jar, if I eat two of them how many do you get.
A: There are a lot of good games here that I've been playing with my 4.5 year old
http://www.mathsisfun.com/games/index.html
A: Do you live in a snowy area?  You could give your child the challenge of figuring out the order in which he or she should put on winter gear.  They usually go for the gloves and boots first, which makes it tough to put on a coat or snow pants.
This isn't as purely mathematical as some of the other suggestions, but it's practical.  From experience, I can tell you that it's hilarious to watch.
A: Roll out a bunch of pennies and show the distinction between primes and composite numbers by asking to make rectangular arrays out of various number of coins. 
A: It's only one small problem, but one of my favourite puzzles as a kid was the following:

You are standing at the edge of a 10 meter pond, and you throw a stone halfway across, spinning it, so it skips another half of the remaining distance. On each bounce, the stone skips one half of the remaining distance to the opposite edge. How many skips does the rock make getting to the other side?

This (probably worded better) was in a giant MENSA book my parents got me (and my twin brother) around our fifth birthday, and although a lot of the puzzles may be somewhat inappropriate, it may be a good source of some reasonable problems to get the gears in the brain going.
My thoughts on the problem, after a puzzled minute or two, was that the stone would, after no number of skips, reach the other side (sure, the stone has an unrealistic width/length/radius/... for this to be true, but just the thinking and realizing half the remaining distance would "never" get to the other side was quite an fun/odd result to come up with),  so it's perhaps not unreasonable to give this to a 3-5 year old as it only really uses the concept of (well, skipping stones, and) "one half" of something. Twenty years later and it's the only problem I remember from the entire book!
A: We have a museum at our university, with an installation of morenaments, an application I wrote myself. We find visitors of all ages spending hours drawing wallpaper ornaments with these. For iOS devices like the iPad, there is a newer development called iOrnament. These applications can be great to use an aesthetic faszination and turn it into curiosity about the governing rules behind the symmetries. With a guiding hand, a lot of structure can be discovered even by the very young. The numbers involved won't exceed 6, unless you want to talk about 360° in a full circle. But to me, math is about structure, not numbers.
A: Here is what seems like a great resource, that I learned of from the Albany Math Circle: http://mathpickle.com/K-12/Videos.html
