I - What day of the week was January 1, 2000?
Step 2 - The Year is 00 so it is already reduced by the largets multiple of 28, and so 00 plus 0 (00 divided by 4) is 0. Add 0 for the Century and subtract 1 because it is a leap year and the year 2000 is divisible by 400, and you get -1 or 6.
Step 3 –Add 1 plus 6 and you get 7 or 0. Looking up the Day of the Week for that result in the Day of Week table we get a Saturday for January 1, 2000.
let's find each of the Month, Day, and Year of DATE, when that one element is missing and the Day of Week is known.
II - In what Month(s) in the Year 2000 was the first of the month a Saturday?
Click here for the basic technique for finding the Month.
Steps 1-3: Going through the three-step basic formula using 0 (?) for the Month, we get 1, that is, the 1 from Step 1 plus the 0 from Step 2 (i.e., 0 + 0 + 0 -?) for a total of 1. (We did not subtract out 1 for the leap year adjustment in January and February because we do not know that it is a January or February.
Step 4:
We know we have a value of 0 for the Saturday and that the result from steps 1-3 is 1, so we need to add 7 to former, and so 7 minus 1 equals 6, which is value of the target Month we are looking for.
Looking at the Month table (remember, in leap years, January and February values are one less than shown on the table, 6 and 2 respectively), we see that the months of January, April and July with a value of 6 qualify for the target Month.
III - What were the various Saturdays in January 2000?
Click here for the basic technique for finding the Day.
Steps 1-3: Going through the three-step basic formula using 0 (the ?) for the Day, we get a 6, that is, 0 from step 1 (we did not know the Day) and -1 from step 2, for a total of -1, which is the same as 6 (7-1=6).
Step 4:
We know from the Day of Week table we have a value of 0 for Saturday and we know that the result from steps 1-3 is a 6, so since we cannot go from a 6 to a 0, we add 7 to the 0, and so 7 - 6 equals 1, which is the first Saturday we are looking for. To find the 2nd, 3rd and 4th Saturday's, add 7, 14, and 21 for the 8th, 15th and 22rd.
IV - In what Year in or after 1998 did January 1 fall on a Saturday?
Click here for the basic technique for finding the Year.
Steps 1-3: Since we do not know the Year we are looking for, we only have to go through step 1 of the three-step basic formula, so the result from that step is 1 (0 + 1 = 1).
Step 4:
We know we have a value of 0 for the Day of the Week, (Saturday) and we know that the result from steps 1-3 is a 1, and because the latter is greater than the former, we add 7 to the former, so 7 minus 1 i2 6, which is the value of the Year (in January) we are looking for.
Since we know the start Year, 1998, when we calculate the value for that year we get 4 (i.e., 98-84=14; 14/4=3; 14+3+1-0=18; 19-14=4). Step forward one year and we get 5 for 1999. Step forward one more year and we land on a leap year (2000) and so we get 6 for January/February and 0 for the rest of the year. We have a January date and so the year 2000 is the year we are looking for.
If we want to find the next year in which January 1st will be a Saturday, let's leap to the pre-leap day value four years from 2000, which is 2004, also a leap year. So add 5 (remember, go forward 4 years but add 5 to the value) to the value and we get 4, that is, 5 + 6 = 11 - 7 =4 . Step forward one year to 2005 for a value of 5, and then one more year to 2006 for a value of 6, and that's the next year January 1 will be on a Saturday.
V- In what Century did January 1 fall on a Saturday in the the year '00?
Click here for the basic technique for finding the Century.
Steps 1 - 3: Going through steps 1 to 3 in the basic formula using 0 (?) for the Century, we get 1 from step 1 and 6 from step 2 (0 + 0 + ? + 1 = -1 or 6) and so 1 plus 6 equals 7 which is he same as 0.
Step 4: We know we have a value of 0 for the Day of the Week, (Saturday and we know that the result from steps 1-3 is also a 0, so 0 ,- 0 equals 0, which is value of the target Century we are looking for.
From the Century table the 21st Century (2000s) have the value of 0 we are looking for, and so the year we are looking for is 2000.
