## Holiday Calculations

###### By Mike O’Hern, Center Director of Mathnasium of West Knoxville

With the holidays coming soon, I naturally start thinking about math.  No, not really.  I start thinking about the birth of Christ, love and charity, family and friends, and Santa Claus, just like you.  But in this space I’m all about the math, so let’s get to it.

Like the twelve days of Christmas, for instance.  How many gifts did I get from my true love?  Remember, one gift on the first day, two gifts on the second, three on the third and so on, for twelve days.  (I think I like this idea!)  Well it’s easy enough to simply add them all up, but what if this went on for a lot longer than twelve days?  We should have some way of figuring that out without having to simply add up all of those numbers.

Imagine that, we do!  Turns out that if you knew the average of all the numbers you could simply multiply that average by the number of numbers in your series and voila – the sum!  Let’s try it.  You go add up one through 12 while I figure out the average value.  That would be (12+1)/2, or 6 1/2 .  Ah!  Just add the first and last number and divide by two to get the average value.  Now 6 1/2 x 12 gives us 78.  Even more exciting is that this will work with any series of numbers, regardless of where it starts or even how big the steps are!  You can have some fun with that!

#### Even more exciting is that this will work with any series of numbers, regardless of where it starts or even how big the steps are!

Or here’s another very useful bit of math for the holidays.  You have a big crowd of friends and loved ones at the house and your dad proposes a toast.  He says a few words (not even inappropriate this time!), everyone raises their glasses and clink!  You, being the best host or hostess ever, want to make sure everyone feels welcome and loved and a part of the group, so you want to make sure everyone clinks their glass with everyone else.  Here’s the question: how many clinks should you hear?

Hard to believe, but we have a way to figure that out, too!  In math we call it combinations, that is, how many different ways to do something.  (No, don’t start squabbling about permutations – we’re keeping this simple.)  Now if I were getting all mathy on you I would start talking about factorials and stuff, but I’m fairly certain that for now we can just boil it all down to one simple equation for this case.  Multiply the total number of people in the group by that number minus one and divide that in two.  So if there are ten people, we have (10 x 9)/2 = 45.

Finally I want to make this as simple as possible so that you can do it in your head really quickly and impress your friends.  Notice that since these are consecutive numbers, one of them will always be even.  So you can cut the even one in half first, then multiply by the other one.  In our previous example, half of 10 is 5, 5 x 9 = 45.  Done.  Say there are 9 people, so (9 x 8)/2 = 36, that is half of 8 is 4 and 4 x 9 = 36.

So there you go!  Go forth and make sure everyone feels welcome and loved and a part of the group this holiday season!