Friday the 13th Project

by Brant Chesser

We want to know how many Friday the 13th's will occur in a given year from 2000 to 2100. We first start out by creating an excel spreadsheet which will allow us to see a pattern if there is any between the years. Also helpful is seeing what day January 1st falls on and how the following days correspond to that day. Here are some spreadsheets to start us off:

January 1st Days

We start to see a pattern forming in the year 2009 that starts over where it began in 1998. This is when three Friday the 13th's occur within the same calendar year. The three months that contain Friday the 13th's are February,March and November. The way this is able to take place is because February and March's Friday's are four weeks apart while March to November will be exactly thirty-five weeks apart when this patten occurs with the distance from February's 13th to November's 13th being thirty-nine weeks in length. New Year's Day will fall on a Thursday when we have three Friday the 13th's in one calendar year. The next time this event should occur is in 2020, but with that being a leap year, it will not take place because February 29 will cause March and November to be pushed back one more day, hence making them Saturday's instead of Friday's. So the next time that a New Year's Day is on a Thursday and will contain three Friday the 13th's will be in 2026 and also in 2037. Thus following the eleven year pattern that we saw between years with three Friday the 13th's between 1998 and 2009. In a non-leap year when February contains a Friday the 13th, March will have to also, because they are exactly four weeks apart to the day. In a leap year, February 29th causes March not to have a Friday the 13th, because they are exactly four weeks and one day apart at that point. When New Year's Day takes place on a Thursday in this eleven year pattern each Friday is the second day in each respective week. So for February, week 6 day 2, for March, week 10 day 2 and for November week 45 day 2.See if you notice the pattern in the spreadsheet below:

Through this investigation we can see that when a leap year takes place that any Friday the 13th after February 29 will take place again the following year. So lets delve a little deeper into the years and see if the pattern keeps continuing that every 11 years or so with New Year's Day on a Thursday that we will have three Friday the 13th's when it does not take place in a leap year.

From the above spreadsheet we can see that in 2043 that New Year's Day falls on a Thursday in a non-leap year. This gives us three Friday the 13th's. So it breaks up the eleven year pattern. It starts back again in 2054 when New Year's is on a Thursday again. The three Friday the 13th's fall in February, March, and November.

Through Mod 7 and dividing the 365 or 366 days of the year into weeks and days we can see that the only way Friday the 13th's can occur in back to back months is in February and March. Take a look at the excel spreadsheet below:

As we can see from above a leap year like the one above on the far left of 2008 is the only way to seperate February and March from being exactly four weeks apart. In 2009 when they are both Friday the 13th's in conesecutive months, February is in the sixth week and the second day while March is in the tenth week and the second day. Being four weeks apart exactly allows each Friday the 13th to fall in back to back months. Lets see the difference in another spreadsheet below:

So when the spreadsheet is worked out we can see that with in Mod 7 each day will remain the same number except for leap years which will just push each day back one number. We can see that September 13th is 364 in Mod 7 while October 13 is 406. September 13th is 365 during leap years while October 13th is 407. From the excel link above you can view that no two other months are exactly four weeks apart and fall on the same day like February and March falls. So we see that the day in October will always be two days ahead of four weeks to the corresponding day in September. So lets see a spreadsheet of all the possible Mod 7 numbers of all 13th days in each month which gives us the following:

So as we investigate the possibility of having two consecutive months that have Friday the 13th's here is our evidence that it can only be in February and March in a non-leap year. February and March in a non-leap year both are the 2nd day of the week exactly four weeks apart, hence 62 and 102. April is on the 5th day or 6th, May on the 7th or 1st, June on the 3rd or 4th, July on the 5th or 6th, August on the 1st or 2nd, September on the 4th or 5th, October on the 6th or 7th, November on the 2nd or 3rd, and December on the 4th or 5th. So no two other consecutive months have the same days of the week. Through this we can see that February and March can be the only two consecutive months with Friday the 13th's. Mod 7 helps us see so much and makes the research investigation easier to spot with patterns and what can and can't happen in non-leap years and also leap years. So now we know when to watch ourselves on each Friday the 13th in the coming years of our lives.

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