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A Maths Problem ...
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... today is something of a 'mirabile die' as my old Latin teacher would have said.
I have two sources of income - one pays every four weeks on a Tuesday, the other pays on the 15th of every month - and today the two days coincided - I am (temporarily) 'in funds' :)
But (as I love playing with numbers) I then started thinking about how often this must happen - a simple spreadsheet produced the previous and next dates (just keep either adding or subtracting 28 days to today's date for a list) ...
But there didn't seem to be an obvious pattern - so I had a very large cup of coffee and attempted a bit more analysis.
I now have the period over which the sequence repeats and an answer for how often the 15th and 'every fourth week' coincide.
Q1 : What is the period for repeats
Q2 : How often do the days coincide in that period
Q3 : Does it matter that it's the 15th?
If that''s too easy (hello jim360) then try
Q4 : What if I was paid on the last calendar day of each month - how often would that coincide with the 'every fourth week' sequence.
Happy Puzzling
SD xx
I have two sources of income - one pays every four weeks on a Tuesday, the other pays on the 15th of every month - and today the two days coincided - I am (temporarily) 'in funds' :)
But (as I love playing with numbers) I then started thinking about how often this must happen - a simple spreadsheet produced the previous and next dates (just keep either adding or subtracting 28 days to today's date for a list) ...
But there didn't seem to be an obvious pattern - so I had a very large cup of coffee and attempted a bit more analysis.
I now have the period over which the sequence repeats and an answer for how often the 15th and 'every fourth week' coincide.
Q1 : What is the period for repeats
Q2 : How often do the days coincide in that period
Q3 : Does it matter that it's the 15th?
If that''s too easy (hello jim360) then try
Q4 : What if I was paid on the last calendar day of each month - how often would that coincide with the 'every fourth week' sequence.
Happy Puzzling
SD xx
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A spreadsheet approach shows that, excluding the August 2017 coincidence, it will happen 5 times in the next 20 years and 14 in the next 50.
But maybe the general problem can be seen by ignoring the actual calendar and looking over a long term. You are paid on the Xth of each month (in your case the 15th). In the long run the probability of the Xth being a Tuesday(*or any other specified day of the week) is 1 in 7. You are also paid in 1 in 4 Tuesdays*. So I think that means that, in the long run, 1 in 28 of your monthly payments will coincide with your 4-weekly payments.
So once every 28 months? I know that gives a shorter cycle (or higher frequency) of actual coincidences than the spreadsheet model but I think that just reflects the current calendar and your cycle- I found another cycle of dates which gave more coincidences of once every 2 years
But maybe the general problem can be seen by ignoring the actual calendar and looking over a long term. You are paid on the Xth of each month (in your case the 15th). In the long run the probability of the Xth being a Tuesday(*or any other specified day of the week) is 1 in 7. You are also paid in 1 in 4 Tuesdays*. So I think that means that, in the long run, 1 in 28 of your monthly payments will coincide with your 4-weekly payments.
So once every 28 months? I know that gives a shorter cycle (or higher frequency) of actual coincidences than the spreadsheet model but I think that just reflects the current calendar and your cycle- I found another cycle of dates which gave more coincidences of once every 2 years
The full (boring) answer is that there is a cycle of 112 years to cover all the possible permutations.
Normally calendar based puzzles repeat every 28 years (to allow for the four year leap year cycle and the seven days of the week), but 28 years doesn't divide exactly by 4 weeks, so you have to go up to 112 years for the full cycle in this case.
There are then 48 coincidences in the 112 years (I'll spare you the arithmetic) for any date between the first and 28th of the month, 45 coincidences for the 29th, 44 for the 30th and 28 for the 31st.
So my payment dates will coincide 48 times in 112 years.
This is an average of 3 times in every 7 years, or (as factor says) once every 28 months - but often it will be much longer than that and occasionally as little as one month.
nerdy dave xx
Normally calendar based puzzles repeat every 28 years (to allow for the four year leap year cycle and the seven days of the week), but 28 years doesn't divide exactly by 4 weeks, so you have to go up to 112 years for the full cycle in this case.
There are then 48 coincidences in the 112 years (I'll spare you the arithmetic) for any date between the first and 28th of the month, 45 coincidences for the 29th, 44 for the 30th and 28 for the 31st.
So my payment dates will coincide 48 times in 112 years.
This is an average of 3 times in every 7 years, or (as factor says) once every 28 months - but often it will be much longer than that and occasionally as little as one month.
nerdy dave xx
NJ you will find the ten hour day etc is the Fransh revolutionary calendar
the period of ten days may have been a Franciade
abandoned before they abandoned messidor etc
FF has almost got it
every 28th day is a Tuesday because you set it as so
so... it boils down to .... how many times a year does the 15th of the month fall on a Tuesday ?
yeah well not very often is the answer
every 28 d means there will be 13 payments in the year and 15th of each month there will be 12
the month with two payments can't possibly a coincider with a 15th because the two dates 28 d apart have to be 1 and 29 or 2 and 30 or possibly 3 and 31
so I rather lost interest afteer that
I like FF's answer
the period of ten days may have been a Franciade
abandoned before they abandoned messidor etc
FF has almost got it
every 28th day is a Tuesday because you set it as so
so... it boils down to .... how many times a year does the 15th of the month fall on a Tuesday ?
yeah well not very often is the answer
every 28 d means there will be 13 payments in the year and 15th of each month there will be 12
the month with two payments can't possibly a coincider with a 15th because the two dates 28 d apart have to be 1 and 29 or 2 and 30 or possibly 3 and 31
so I rather lost interest afteer that
I like FF's answer
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