So, have you broken your New Year’s resolutions yet? While most people ring in the new year with resolutions (loose 10 pounds, clean out the closet, eat healthy), mathematicians like to talk about the features of the numeric value of the new year. 2016 is a fun number, not only because it is divisible by 4 which makes this a leap year*, but because 2016 is a TRIANGULAR NUMBER.

You might have heard of the SQUARE NUMBERS: 1, 4, 9, 16, 25, 36, … All of these numbers can be thought of as dots and then arranged into squares like so:

But do you know about the TRIANGULAR NUMBERS? TRIANGULAR NUMBERS: 1, 3, 6, 10, 15, 21, … can be thought of as dots as well and arranged into triangles like so:

If we carried this pattern out a few more times (ok a lot more times), we would come upon 2016. 2016 is the 63rd triangular number! I leave the dot drawing to you and your students…

* Not every year divisible by four is a leap year, but that is the subject of another blog post.

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To explain the leap year exception. Only centuries divisible by 400 are leap years. 2000 was. 2100 won’t be.

Stolen from the internet:

“The exact length of a solar year is actually 11 minutes and 14 seconds less than 365 ¼ days. That means that even if you add a leap day every four years, the calendar would still overshoot the solar year by a little bit—11 minutes and 14 seconds per year. These minutes and seconds really start to add up: after 128 years, the calendar would gain an entire extra Sat. So, the leap year rule, “add a leap year every four years” was a good rule, but not good enough!”

So why are centuries not divisible by 400 not leap years? E.g. 2000 was a leap year but 1900 wasn’t and 2100 won’t be.

Stolen from the internet:

“The exact length of a solar year is actually 11 minutes and 14 seconds less than 365 ¼ days. That means that even if you add a leap day every four years, the calendar would still overshoot the solar year by a little bit—11 minutes and 14 seconds per year. These minutes and seconds really start to add up: after 128 years, the calendar would gain an entire extra Sat. So, the leap year rule, “add a leap year every four years” was a good rule, but not good enough!”

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