Question

Dear Mitch,

I believe I read somewhere that in addition to the very challenging Christmas problem you posted last week you would post some of medium difficulty levels and then, closer to the holiday, some even easier ones. I would really appreciate it if you would do that.

Sincerely,

Mr. Michael Wood

High School Teacher

near Haddam, CT

Answer

Dear Mr. Wood,

In order to do the best he can to get every present to each and every boy and girl in the world before the sun came up on Christmas day, and before tiring the reindeer out with too many unnecessary moves in the sky, every year Santa Claus considers hundreds of different paths for his sleigh and reindeer. And, although he does own a calculator, he (not surprisingly) takes a very traditional approach and prefers to work all the numbers out with paper and pencil. Of course he then checks over his work twice to make sure he hadn't made any naughty mistakes, careless, sometimes thrice to be nice,

Like many serious learners, he doesn't wait until the last minute but is already hard at work with his problem solving. And as he had learned long ago, a logical step-by-step approach seemed the best way to arrive at successful solutions.

So, on an evening not too long ago, he was sketching out a path on his new electronic white board and paused to study his diagram. It was a triangle. Like this:

And there were two pieces of information he needed to consider:

1) He had promised his reindeer that he would do all he could to avoid sharp turns (which were the kind of turns the heavier reindeer did not appreciate);

And,

2) By tracing, folding, and retracing various parts of his diagram he discovered that **angle x + angle y** was ** equal** to

**angle z**. (Although here the diagram is no longer drawn to scale, as I was unable to obtain it before it was covered in snow, ice, and stepped on by, well, by a lot of little people rushing into the workshop to get an early start. So it is distorted and now must be redrawn by anyone who feels he or she needs a diagram to help Santa, though Santa has already realized that the mathematics really does not require a diagram to figure the mystery out.

Now he wanted to know if there was a way to figure out the

measurement of angle z, and, if so, what was that angle's

measurement?

One of the elves overheard Santa's grumbling and slipped a note under his

office door. It said: "I think you can figure out the measurement of angle z,

and I think that it is 180 degrees ** or** 90 degrees

**60 degrees**

*or***45**

*or* degrees ** or** 30 degrees."

As it turns out the Elf was correct. Now all that has to be done is figure

out which of his ideas is the correct one and how Santa can be sure

about it.

Hope this helps get students thinking and having fun,

Mitch