I - What day of the week was February 1, 1945?
Step 1 - Value for February from the Month Table is 3 so add 3 to 1 and you get 4. (You do not need to subtract a multiple of 7 because the number already is between 0 and 6.) Hold this number to step 3.
Step 2 - Subtract 28 from 45 and you get 17. Now divide the 17 by 4 and you get 4 plus some decimal, which you can drop. Add the 17 and 4 together and you get 21 (which you can drop immediately now because it is a multiple of 7, but we will not here) . Add to the 21 the Century value of 1 from the Century Table and you get 22. The month is February but since there was no decimal in the above calculation the year us not a leap year so we do not have to subtract 1 from this, meaning we are left with 22.
Step 3 - Add the 4 from step 1 to the 22 from step 22 and you get 26. Subtract the largest multiple of 7 contained in 26 (i.e., 21) and you get 5. Looking up 5 in the Day of Week Table you get THURSDAY. February 1, 1945, was a Thursday.
Now let's to find each of the Month, Day, and Year of February 1, 1945, when that one element is missing and the Day of Week is known.
II - In what Month(s) in 1945 was the 1st on a Thursday?
Steps 1-3: Going through the three-step basic formula using 0 (the ?) for the Month, we get 2, i.e., (? +1 = 1) + (17+ 4 + 1 -0 = 22+1 = 23 - 21 =2).
Step 4:
Since we know we have a Thursday as the Day of Week, that's a 5. We subtract the 2 from steps 1-3 from the 5 (we do not have to add 7 to the 5 since it is greater than the DOW we are looking for) and we get 3. This is the value of the Month we are looking for.
Looking at the month table (remember, in leap years, January and February are one less than shown on the table), we see that the months February, March and November have a value of 3, so the month we are looking for is one of those.
III - On what Day did the first Thursday in February 1945 fall?
Steps 1-3: Going through the three-step basic formula using 0 (the ?) for the Day, we get 4 (i.e., 3 + ? = 3) + (17+ 4 + 1 -0 = 22) and 22 + 3 = 25 - 21 = 4).
Step 4:
Since we know we have a Thursday as the Day of Week, that's 5. Subtract the 4 ( from steps 1-3) from the 5 and you get 1, which is the reduced value of the Day we are looking for.
The first target Day (a Thursday) of that month we are looking for is the 1, the 1st. Add 7, 14, and 21 to the 1st to get the 2nd, 3rd and 4th Thursdays and you get the February 8th, 15th, and 22nd.
IV - In what Year or years (in or after 1940) did February 1, 1945, fall on a Thursday?
Steps 1-3: Going through the three-step basic formula using 0 for the full year (century plus year), we get 4 (i.e., 3 + 1 = 4).
Step 4:
Since we know we have a Thursday as the Day of Week, that DOW has a value of 5. Since 4 is less than 5, we simply subtract the 4 from the 5 to find the value of the full Year (century plus year) we are looking for, and we get 1. Remember that because the date we are looking for is in February, this is the value of the full Year for a non-leap year, so whatever value for a full year we derive, we must subtract 1 for January and February dates in leap years.
To get to this answer to this example, we need to have some year to start with, but preferably not 5 years beyond what might be the year we are looking for. Let's say in this case that 1940 seems to be a safe year to start with. So, Calculating the value for 1940, we get 1 as follows: (40-28 = 12); 12/4=3; 12+3+1=16; 16-14=2. So the value for the full Year is 2, but since it is a January and February date, we need to subtract, and we get 1, which is one of the possible years we are looking for. I always give at least two possible years, and so I would continue with my calculations. You must continue with your calculating if the first year you started with was not a target year.
There are several ways to move forward. The first was is to move forward one year at a time, calculating the year value for each each time, just as I did with 1940 above, until you find the first year or the next possible year you are looking for, but that would be time consuming. Let's call this the "yearly calculation" method.
The third was is to compute the value for some starting year, as we did above for 1940, and then jump forward four years by adding a value of 5 to what we got for the start year. For 1940 we get a 1/2 (it was a leap year), so for 1944 (a leap year) we get a 6/7 or 6/0. We are still short so we need to move forward one year to 1945 and we add 1 to the 0 alone (1945 is not a leap year) and we get 1, the value we are looking for. So 1940 and 1945 are two target years. Let's see what the next year is after 1945 by this method. Add 4 to the year (1945+4=1949) and 5 to the value (1+5=6) and we get a value of 6 for 1949. Step forward one year to 1950 and we get a 0; step forward another year to 1951 and we get 1, the next target year. So, 1940, 1945 and 1951 all had February 1 fall on a Thursday.
V - In what Century was February 1st a Thursday?
Steps 1 - 3: Going through steps 1 to 3 in the basic formula using 0 (?) for the Century, we get a 4 (3+1=4; 45-28=17; 17/4=4; 17+4+?-0=21; 4+21=25; 25-21=4).
Step 4:
We know the value for the Day of the week we are looking for (a Thursday) is 5, so 5 minus 4 = 1, the value of the Century (from the table) we are looking for.
Looking at the Century table, we see that the value for the 20th century (1900s) is 1, so the year we are looking for is 1945.
