Problem of the Week Archive

Triskaidekaphobia – November 9, 2015

Problems & Solutions Triskaidekaphobia is an irrational fear of the number 13. Because of fear and superstition surrounding the number 13, it is common in tall buildings for the 13th floor to be skipped when installing elevators buttons, it is instead numbered 14. A particularly fearful hotel owner decided to skip floor 13, as well as, any multiples of 13 when numbering the floors. He did the same thing when numbering the rooms. If each floor has rooms numbered to 100 and the elevator has buttons numbered to 30, how many rooms does the hotel have?

The multiples of 13 between 1 and 100 are 13, 26, 39, 52, 65, 78 and 91, seven multiples. If the hotel has rooms numbered up to 100 but skips multiples of 13, this means that each floor has 100 − 7 = 93 rooms. The floors are numbered up to 30, but there are two multiples of 13 between 1 and 30, so there must actually be 30 − 2 = 28 floors. This means the total number of rooms in the entire hotel is 93 × 28 = 2604 rooms.

Friday the 13th is a day that is believed widely to be unlucky, a superstition thought to be among the most common in the United States. This week, the 13th will fall on a Friday. In what month will the next Friday the 13th occur?

There are 30 days in the month of November, therefore 30 days from November 13th will be December 13th. A full week is 7 days, and since 30 days is not divisible by 7, we know December 13th will not be on a Friday. If we divide 30 days by 7 days/week, we get 4 weeks with a remainder of 2 days. This means December 13th will occur 2 days of the week past Friday, on Sunday. The month of December has 31 days, giving us 4 full weeks and a remainder of 3 days. January 13th will therefore occur 3 days of the week later than December 13th, on a Wednesday. January also has 31 days, so February 13th will occur 3 days of the week later than January 13th on a Saturday. February has 29 days because it will be a leap year, so this gives us 4 weeks and a remainder of 1 day, making March 13th a Sunday. March has 31 days, putting April 13th on a Wednesday. April has 30 days, putting May 13th on a Friday. So we now know that the next Friday the 13th will occur in May of 2016.

What is the maximum number of Friday the 13ths that can occur in one calendar year?

From the previous problem, we know that months with 31 days have 4 full weeks and 3 days and months with 30 days have 4 full weeks and 2 days. February has 28 days or 4 full weeks, except on a leap year, when it has 29 days or 4 full weeks and 1 day. Let’s assign the days of the week the letters A, B, C, D, E, F and G, we do not need to specify which day of the week corresponds to which letter. In January if an arbitray seven numerical day period corresponds to the days of the week ABCDEFG, in that order, then next month that same seven numerical day period will correspond to days of the week shifted by 3 days, since there are 31 days in January. So in February, those same seven days, will be the days of the week EFGABCD. For example, if we pick January 6, 7, 8, 9, 10, 11, 12 as our seven numerical days and assume they correspond to Sunday through Saturday, then February 6, 7, 8, 9, 10, 11, 12 will correspond to Thursday through Wednesday. If we are looking at a non-leap year, then in March those days will be the same, EFGABCD. March has 31 days so in April those days will shift by 3 and be BCDEFGA. Continuing this for all of the months, based on the number of days in each month, we get the following pattern for the days of the week: January ABCDEFG May GABCDEF September CDEFGAB February EFGABCD June DEFGABC October ABCDEFG March EFGABCD July BCDEFGA November EFGABCD April BCDEFGA August FGABCDE December CDEFGAB Looking at the first letter for the specific seven numerical day period each month, we see the sequence is A, E, E, B, G, D, B, F, C, A, E, C. Here the letter that occurs the most is E and it occurs three times. If E is Friday and we assign the first day in the seven numerical day sequence to be the 13th, then Friday the 13th would occur 3 times. The same conclusion can be drawn looking at the second, third, fourth, fifth, sixth or seventh letter in the sequence for each month – there will always be a letter occuring 3 times. If you repeat this process for a leap year, you will again see that, in one calendar year, Friday the 13th can occur a maximum of 3 times.