I - What Day of the Week was January 11, 1988?
Step 1 - Value for January from the Month Table is 0 so add 0 to 11 and you get 11. Subtract 7 and you get 4. Hold this to step 3.
Step 2 - 88 minus 84 is 4. Take the 4 and and add to it the number you get by dividing it by 4 (you get 1.0) and dropping the decimal (there is no decimal in this case) and you get 5 (i.e., 4 plus 1). Add to the 5 the value of 1 you get from the 1900's (century) from the Century Table and you get 6. But since this date is both a leap year (no decimal when you divided by 4) and in January or February, you must subtract 1 which makes the number 5 (6-1=5).
Step 3 - Add the 5 from step 2 to the 4 from step 1 and you get 9. Subtract the largest multiple of 7 contained in 9 (i.e., 7) and you get 2. Looking up 2 in the Day of Week Table you get MONDAY. January 11, 1988 was a Monday.
Now let's find each of the Month, Day, and Year of January 11, 1988, when that one element is missing and the Day of Week is known.
II - In what Month(s) in 1988 was the 11th on a Monday?
Steps 1-3: Going through the three-step basic formula using 0 for the Month, we get 17 (i.e., 0 +11 + 4 + 1 + 1 -?), which means 3. (The question mark is for the leap year adjustment, which is the same as 0.) Since we do not know the month, we do not know if we have to subtract 1, so the result from steps 1-3 is 3.)
Step 4:
We have 3 from steps 1-3, and we know from the table that Monday is the day of the target date, but since 3 is greater than 2 we have to add 7 to the 2 for 9, so 9 minus 3 equals 6, which is the value of Month we are looking for.
We know from the table that April and July have a value of 6 but we also know that since 1988 was a leap year, the value for January and February must be reduced by 1, which means for January the value is not 0 but 6. So, in 1988, the months when 11th of the month fell on a Monday were January, April and July.
Step 4:
We have 3 from steps 1-3, and we know from the table that Monday is the day of the target date, but since 3 is greater than 2 we have to add 7 to the 2 for 9, so 9 minus 3 equals 6, which is the value of Month we are looking for.
We know from the table that April and July have a value of 6 but we also know that since 1988 was a leap year, the value for January and February must be reduced by 1, which means for January the value is not 0 but 6. So, in 1988, the months when 11th of the month fell on a Monday were January, April and July.
III - On what Day did the second Monday in January 1988 fall?
Steps 1-3: Going through the three-step basic formula using 0 for the Day, we get 5 (i.e., 0 + 0 + 4 + 1 + 1 -1). Since we know we have a Monday as the Day of Week, that's 2.
Step 4:
We have a 2 from steps 1-3 and we know that the Day of the Week is a 5. But since 5 is greater than 2, we must add 7 to the 2 for 9. So 9 - 5 is 4.
The 4 is the first Monday of that Month, but we want the second, so we add 7 for 11. January 11, 1988, was a Monday. (The Monday's in that month were 4, 11, 18, and 25.)
IV - In what Year did January 11, 1988, fall on a Monday?
Steps 1 -3: Going though steps 1 to 3 for the month and day only we get a 4, that is, 0 + 11 = 11 - 7 = 4.
Step 4:
We have the 4 from Steps 1-3 above and we know the Day of the Week we are looking for is a Monday, which is a 2. Since the 4 is greater than the 2, we must add 7 to the 2 to get 9, and so 9 minus 4 equals 5, which is the value of the year we are looking for (remember, in a leap year, so the value for the full year mist be reduce by 1).
Steps 1 -3: Going though steps 1 to 3 for the month and day only we get a 4, that is, 0 + 11 = 11 - 7 = 4.
Step 4:
We have the 4 from Steps 1-3 above and we know the Day of the Week we are looking for is a Monday, which is a 2. Since the 4 is greater than the 2, we must add 7 to the 2 to get 9, and so 9 minus 4 equals 5, which is the value of the year we are looking for (remember, in a leap year, so the value for the full year mist be reduce by 1).
The next thing we do is to look for the year or years with the value we are looking for (a 5) starting with some year that is the most likely earliest year it could be. In this case, let's say we determined that the earliest year could be 1984. Computing the year value (step 2 in the basic formula) only for 1984 we get 0 (84 - 84 = 0 and 0/4 = 0; so 0 + 0 + 1 for the century and - 1 for the adjustment for a January/February in a leap year). Since we are looking or a 5, we must move on to a future year. Of the three methods we can use to go to the year value we are looking for, since we the gap in vales between 1984 and 1988 is large a (5, computed as 5 - 0), let's jump forward 4 years to 1988 and add 5 to the value of 0 we have for 1984 and we get a 5, which is precisely what we are looking for. So the first year our target date could be is 1988.
If you want to get the next possible year just in case, let's jump another 4 years to 1992 and add 5 to our value of 5 for a value of 10, which is 3 (10 - 7 = 3). That's not the value we are looking for so let's walk forward on year to 1993 and add 1 to the value of 3 that we have for 1992 and we get 4. Nope, that's not good either, so let's step forward one more year to 1994 and add 1 to the value of 4 we have for 1994 and we get 5, which is the value we are looking for, so the next valid year is 1994. So 1988 and 1994 are the first two years after 1984 with the value we are looking for.
V - In what Century was January 11th in the year '88 a Monday?
Steps 1 - 3: Going through the three-step basic formula for all the things we know, but using 0 (the ?) for the Century, we get 1 (i.e., 0 + 11 + 4 + 1 +? -1 = 15 minus 14 = 1).
Step 4:
We know we have a Monday as the Day of Week, which is 2, and we know we have a 1 from Steps 1-3, so 2 minus 1 equals 1. This 1 is the value of the Century we are looking for.
All we need to do at this point is to look at the Century table, and we see that the 1900's had a 1 value, so 1988 is the year we are looking for.
Step 4:
We know we have a Monday as the Day of Week, which is 2, and we know we have a 1 from Steps 1-3, so 2 minus 1 equals 1. This 1 is the value of the Century we are looking for.
All we need to do at this point is to look at the Century table, and we see that the 1900's had a 1 value, so 1988 is the year we are looking for.
