Dimensional analysis would really help here.
When multiplying or dividing quantities with units, keep the units in the equation to see what you end up with.
For example, the units of speed are distance/time, for example km/h or miles/h or meters/second. If you do math and you end up with meters^2/second or meters/dollar or something, you didn't calculate a speed.
Let me look at Napkin #5 to illustate this:
Premise:
- A Solar Day happens every 24 hours
- A Year happens every 365 days
365 days / 2 = 185.5 days in six months
Now, we need to fit 24 into 185.5
185.5 days divided by 24 = 7.729
Conclusion: 24 does not fit cleanly into 185.5. If it did, it would be a whole number.
Conclusion: The day does not fit neatly into half of the year. Solar Noon will not be at the same place at that point.
Let me rephrase that with units.
Premise:
- A Solar Day happens every 24 hours. 1 solar day = 24 hours. To get a unitless 1, which is always legal to multiply by, take the ratio. 1 = 1 solar day/24 hours
- A Year happens every 365 days 1 year = 365 days. To get a unitless 1, 1 = 365 days / 1 year
365 days / 2 = 185.5 days in six months
0.5 years is what you're looking for, so you can take 0.5 years * 365 days/year. Years cancel, and you get 185.5 days.
THIS CALCULATION IS CORRECT
Now, we need to fit 24 into 185.5
185.5 days divided by 24 = 7.729
185.5 days / (24 hours/day) = 7.729 days^2/hour
THIS CALCULATION IS MEANINGLESS - your units don't make any sense.