So w = 1, x = 1, y = 8, z = 14

]]>28w + 29x + 30y + 31z = 731 ]]>

with a nicely unexpected source for the answer.

At first I felt too lazy to bother.

Then I realized what these numbers were. Ha ha!

Nicest thing for my laziness I don’t even need to figure out the precise values of x y z as you merely ask whether there is any solution 🙂 A perfect fit.

I must come back here more often though – great place. ]]>

And well, yes. The other solution is (x,y,z) = (0,6,6)

This solution was the first adopted by the Romans and provided for

for normal years (x,y,z) = (1,5,6). It was the solution suggested by

Sosigene astronomer of Alexandria in Egypt and adopted by Julius Caesar in 46 a.c. It provided for 31-day January,

February of 29 days, and from March onwards an alternation of months from 31 days followed by months of 30 days. It was also decided to call August the month after July in honor of Augustus Emperor. It was also decided to change the duration of August from 30 days to 31 days at the same time changing the duration of February from 29 to 28 days in normal years and from 30 to 29 in leap years. It was also decided to reverse the duration of the following months from September to December to avoid three consecutive 31-month months.

The calendar as we know has been in force since 8 a.c. until the reform of 1582 Gregorian calendar. ]]>

1 month with 29 days = 29

4 months with 30 days. 30*4 = 120

7 months with 31 days. 31*7 = 217

Total = 366

so (x,y,z) = (1,4,7) ]]>

I first begin using algebraic manipulations to solve the problems,

and the i notice, 29,30,31 and see that these guys are the number of days in the moths,

so after that I see the Calendar to check the number of days of February in 2019, see 28 days, so the equation can’t be solve in the natural numbers.