Thanks for the words of support. Claudia asked for the knitting "pattern" so here goes ....
First off, I've never learned to knit small things, like socks, "in the round" (Those THREE needles!! Without ends!! Yikes!!) but if you do know how to knit socks, doing these in the round would do away with the need for a back seam and would make a much nicer hat!
What you are aiming for (doing it my way = making a flat rectangle) for a doll like Ann Estelle (left above, ca 7.25" head) or Darlene (right above, ca 7" head) is a rectangle that is 8.25"-8.5" long and 3.25" wide. These measurements depend on your yarn weight. The following instructions are for #3 weight yarn (the two solid pink hats use this - from LyonBrand "BonBons" collections.) Lighter weight yarn ("sock yarn") is better, but #3 was what I had on hand in pink, so that's what I used.
Caste on 56 stitches on #2 needles. Knit the rest on #3 needles.
Knit 4 rows of knit/purl/knit/purl ribbing.
Row 5 = knit all the way across
Row 6 = purl all the way across.
repeat 5 and 6 until you have 26 rows total. Bind off.
Sew the two ends together. Then flatten the resulting cylinder, with the back seam falling half way across, and sew the top edges together, from side to side.
I used #4 weight yarn for the middle hat on Sudha, who has a smaller head (ca 6.5"). I made a rectangle about 6.75" long by 2.75" wide by casting on 52 stitches to start with and knitting 22 rows total.
My, what a sophisticated and complicated "pattern" that was, eh??!!! Enjoy. R.