Wooden sandbox on a concrete patio with no bottom.  How do I keep sand from spilling out the bottom? I still want water to drain.?

If the joints of the wooden sides are not too tight, water will leak out anyway.

You could just drill some 3" roughly holes in the bottom and into the soil for drainage. Sand will fill the holes initially and water will leak out.

Or just have a sandbox with no bottom in it.

you don't.  sand will go out with the water that drains out.  the solution to this is a cover that keeps water out when the sandbox is not being used ... which, from a kid's viewpoint, is simply a problem to be discarded.  alternative -- put the sandbox off the patio on the garden [lawn] -- i guarantee that the kids won't care

The walls will doubtless not fit closely with the  floor of the patio, there will be an almost constant gap between floor and wall through which the sand can spill easily, especially with the help of draining water.

You have some narrow boards capping the walls part-way around.  If you did the same at the foot of the walls (all the way around), the sand would find it more difficult to spill the full width of the bottom board.  The weight of the sand on the footboards would help to anchor the structure, and any sand which crept under the inner edge would get trapped there and form a sort of seal which prevented further migration.  

If you felt it necessary, you could pipe a bathroom sealant around the inner edges of the footboards to form a positive seal (leave some gaps for water drainage).  It would peel away easily if you had occasion to move the pit at a later date.

Use weed mat as bottom liner and bring it up the walls 2-3 inches. 

Cover with sand.

 The weight of the sand should keep it in place. 

Or use silicone caulk to adhere it to the wood