# How do I convince my grandmother?

How do I convince my grandmother (who really hates mathematics) that there exist three positive integers $x,y,z$ that satisfy the equation $28x+30y+31z=3650$?

• While it is an interesting question (and I like Gerhard's answer), I wouldn't say this is a puzzle. Commented Feb 1, 2015 at 14:49
• @BmyGuest, having seen the answer I'd say it is a puzzle. If it were just a plain vanilla maths question then it wouldn't be.
– A E
Commented Feb 1, 2015 at 19:26
• You could just tell her the values of x, y, and z. If that doesn't convince her, nothing else will. Commented Feb 1, 2015 at 21:33
• If she hates maths that much, she probably doesn't care enough to need convincing. Commented Feb 2, 2015 at 0:09
• I'm voting to close this question as off-topic because it really is straightforward math. Commented Feb 2, 2015 at 15:34

Take ten ordinary years. They contain exactly 3650 days, and these days are split among several months with 28, 30, and 31 days.

• Ten ordinary years do not contain exactly 3650 days. They contain either 3652 or 3653, depending on which ten consecutive years you picked, due to leap years. The actual length of a solar year is not exactly 365 days, it is approximately 365.24 days. Commented Feb 2, 2015 at 22:08
• @MatthewNajmon I think the ordinary part means non-leap year... Commented Feb 2, 2015 at 22:10
• @MatthewNajmon They said nothing of consecutive years. Commented Feb 3, 2015 at 6:03
• @QuyNguyen2013 Well, then picking out 10 non-consecutive years, your chances of at least one leap-year are high unless you specifically avoid them. If you are intentionally sorting against specific types of year, then the result doesn't really constitute an "ordinary" selection of years. Leap years are not some rare occurrence that can be discounted as outliers. They make up a very significant part (roughly a quarter) of the total population of years, more than enough for them to be relevant in defining what characteristics are "normal" or "ordinary" for a year to have. Commented Feb 3, 2015 at 15:57
• The actual exact time taken for a solar year is utterly irrelevant. This is a Maths problem, not a physical one. In mathematics, we deal with abstractions. The only property of the year we are exploiting is that it is well known even by grandmothers that 365 is a liners combination of 28, 30 and 31. The only reason we are giving the example of the year is that it's easier to understand this particular number theoretic property of 365. The actual amount of time in a solar year doesn't interest us. Commented Mar 20, 2016 at 11:40