Everyone knows Friday the 13th is regarded as a day of bad luck.
Why does every year have at least one of this bad day?
Source: 2010 ISI B Math UGB
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Sign up to join this communityEveryone knows Friday the 13th is regarded as a day of bad luck.
Why does every year have at least one of this bad day?
Source: 2010 ISI B Math UGB
A month has a Friday 13th if and only if it begins on a Sunday.
On a regular (non-leap) year, if January begins on day $k$, $0\leq k\leq 6$ (with $k=0$ being Sunday), then we have that:
With these, we already have day $k$, $k+1$, $k+2$, $k+3$, $k+4$, $k+5$, and $k+6$, so at least one of these months will begin on Sunday, guaranteeing at least one Friday 13th.
For Leap years, the analysis is similar, except that:
So at the latest, you will have a Friday 13th by October.
In a common year the 13ths of the month fall at intervals of $31,28,31,30,31,30,31,31,30,31$,and $30$ days, which reduce mod $7$ to $3,0,3,2,3,2,3,3,2,3$, and $2$. The partial sums mod $7$ are $3,3,6,1$, $4,6,2,5,0,3$,and $5$; since these include a complete residue system mod $7$, at least one of those 13ths must fall on a Friday.
In a leap year the corresponding numbers are $31,29,31,30,31,30,31,31,30,31$,and $30$ for the intervals, $3,1,3,2,3,2,3,3,2,3$, and $2$ for their reductions mod $7$, and $3,4,0,2,5,0,3,6,1,4$, and $6$ for the partial sums mod $7$; once again the partial sums include a complete residue system mod $7$, and one of the 13ths must fall on a Friday.
This is not an elegant method, and maybe there is none, but it works. There are 14 possible one-year calendars:
the common year beginning on Sunday, the common year beginning on Monday, the common year beginning on Tuesday, and so on....
and then the leap year beginning on Sunday, the leap year beginning on Monday, and so on....
Look at each of the 14 calendars, and count the number of Friday the 13ths.
You may prefer to focus only on the months May through November inclusive, and show that the 13ths of those seven months fall on seven different days of the week.
Working mod 7, and letting May 13th = X,
May 13th = X
June 13th = X + 31 = X + 3
July 13th = X + 31 + 30 = X + 5
August 13th = X + 31 + 30 + 31 = X + 1
September 13th = X + 31 + 30 + 31 + 31 = X + 4
October 13th = X + 31 + 30 + 31 + 31 + 30 = X + 6
November 13th = X + 31 + 30 + 31 + 31 + 30 + 31 = X + 2
.The maximum number of days in a month is 31 .The number of days of a week is 7 .The minimum number of days in a year is 315 .31*7=217 .217 means there is all kind of combination of days and weeks .217<315 .Thus there is always Friday the 13th every year, not just Friday the 13th but all kind of dates as well.
This is probably the most easiest way to explain this :D
Problem: Prove that all years have either 1, 2 or 3 months with a Friday The 13th.
Solution:
Case 1: Non leap years
Divide the number of days in each month by 7 and write down the remainder
next to each month as follows:
Jan. 3
Feb. 0
Mar. 3
Apr. 2
May 3
Jun. 2
Jul. 3
Aug. 3
Sep. 2
Oct. 3
Nov. 2
Dec. 3
A month has a Friday The 13th if and only if the first of the month falls on Sunday.
Our task is to record the month(s) having a Friday The 13th for each day of the week
January 1 falls on.
Define Sunday as the first day of the week and Saturday as the last day of the week.
Next to each day of the week write down the difference in number of days
between each day of the week and Sunday.
Then, corresponding to each day of the week write down the month(s) having a
Friday The 13th by arriving at 0 or a multiple of 7 when adding this difference
in number of days to the accumulated monthly remainders from above.
Sun. 0 Jan., Oct.
Mon. 1 Apr., Jul.
Tue. 2 Sep., Dec.
Wed. 3 Jun.
Thu. 4 Feb., Mar., Nov.
Fri. 5 Aug.
Sat. 6 May
Case 2: Leap years
Divide the number of days in each month by 7 and write down the remainder
next to each month as follows:
Jan. 3
Feb. 1
Mar. 3
Apr. 2
May 3
Jun. 2
Jul. 3
Aug. 3
Sep. 2
Oct. 3
Nov. 2
Dec. 3
A month has a Friday The 13th if and only if the first of the month falls on Sunday.
Our task is to record the month(s) having a Friday The 13th for each day of the week
January 1 falls on.
Define Sunday as the first day of the week and Saturday as the last day of the week.
Next to each day of the week write down the difference in number of days
between each day of the week and Sunday.
Then, corresponding to each day of the week write down the month(s) having a
Friday The 13th by arriving at 0 or a multiple of 7 when adding this difference
in number of days to the accumulated monthly remainders from above.
Sun. 0 Jan., Apr., Jul.
Mon. 1 Sep., Dec.
Tue. 2 Jun.
Wed. 3 Mar., Nov.
Thu. 4 Feb., Aug.
Fri. 5 May.
Sat. 6 Oct.