Nooniewaters, I'm a little confused by the whole thing you have going here. Aran yarn should be a little thicker than worsted so supposedly with the worsted you would need more stitches not less.
First puzzle. You are trying to get 18sts=4inches worked over Stockinette. Most worsted should have no trouble getting that gauge and still look pretty good. Puzzle #2. Then you said you were using a 3mm needle to get gauge. I don't have a needle that size, but it is slightly larger than a US2, smaller than a US3. That is a pretty small needle to work worsted on (not that I haven't done it- I knit some socks in worsted on a US 3, but I got 25sts=4inches) and get a gauge that is bigger than 18sts=4inches, that would require you to use fewer stitches than called for.
Things don't add up for me. How many stitches are you getting over 4 inches on the needles you think you want to use?
I wonder what yarn you are trying to use for your hat? I looked the Sublime cashmere merino silk Aran up and its suggested gauge is 18 sts=4inches on an 8, so this hat is right on gauge for this yarn, and not trying to work it particularly tight or loose. What is the suggested gauge for your yarn?
Sue did her figures based on the 74 stitches you think you need, compared to the 84 given in your pattern. If you divide 74 by 84 you get about .88 or 88%. She then multiplied the 59 sts your pattern said to increase by .88 and got 51.92 which she rounded to 52 stitches as the number you needed to increase to keep things in the same proportion as the original hat. She then added your cast on to the number increased: 74+52 and got 126. Since your pattern is a multiple of 11 you either have to go with 121 or 132 which are the two nearest multiples of 11. 11X11=121 and 11X12=132. I'm not good at math either, but since Sue figured it out I was able to figure out how she got those numbers. I try to avoid the "math heavy" fix to a problem whenever I can.
Sue and I are both wondering about your yarn and the stitch count you are getting with it. That would really help to solve the puzzles.