Solution:
No. Every calendar year has at least one month where the 3rd Saturday is the same weekend as the 4th Sunday.
The 3rd Saturday is the same weekend as the 4th Sunday if, and only if, the first of that month is a Sunday. So the original question is equivalent to "Does any calendar year have no month where the first is a Sunday?" But as we will see, in a given calendar year every Day of the Week (DOW) is the first of one or more months.
The science of finding the DOW some number of days from now is familiar to us all. We simply subtract multiples of 7 until we get a remainder less than 7. That is the only math we will need.
January is 31 days long, so whatever DOW New Year's Day is, February 1's DOW is 3 days later, e.g. if New Year's Day is Friday, February 1st is Monday. More generally, if New Year's Day is DOW "X," February 1 is DOW "X+3." For a non-Leap Year:
Non-Leap Year
| Month |
Number of Days |
Days Less Multiple of 7 |
DOW |
| 31 |
3 |
X |
| 28 |
0 |
X+3 |
| 31 |
3 |
X+3 |
| 30 |
2 |
X+6 |
| 31 |
3 |
X+1 |
| 30 |
2 |
X+4 |
| 31 |
3 |
X+6 |
| 31 |
3 |
X+2 |
| 30 |
2 |
X+5 |
| 31 |
3 |
X |
| 30 |
2 |
X+3 |
| 31 |
3 |
X+5 |
By September of a non-Leap Year, each DOW has been the first of some month, i.e. X, X+1, X+2, X+3, X+4, X+5, X+6 have all been the first of some month.
For a Leap Year:
Leap Year
| Month |
Number of Days |
Days Less Multiple of 7 |
DOW |
| 31 |
3 |
X |
| 29 |
1 |
X+3 |
| 31 |
3 |
X+4 |
| 30 |
2 |
X |
| 31 |
3 |
X+2 |
| 30 |
2 |
X+5 |
| 31 |
3 |
X |
| 31 |
3 |
X+3 |
| 30 |
2 |
X+6 |
| 31 |
3 |
X+1 |
| 30 |
2 |
X+4 |
| 31 |
3 |
X+6 |
By October of a Leap Year, each DOW has been the first of some month.
So for any year, Sunday will be the first of some month and hence that month Ed's volunteer commitments will both occur in the same weekend.