Question

Dear Mitch,

I think people would very much appreciate it if VERY SOON you gave more hints on the recent holiday problems you've posted. They really are surprisingly challenging, even compared to other difficult problems you've posted in the past.

Sincerely,

Marissa Derosa

Arnprior, Ontario

Answer

Dear Marissa Derosa,

You are right; it is time for more hints.

First, the triangle question, in which Santa is choosing a route...

A lot of people guess that the vertex of the triangle being asked about is ninety degrees, because it kind of looks like that, BUT you can never assume that from a diagram unless you are given other information that indicates a ninety degree angle. (For example, if it states that the triangle was formed by cutting a square along its diagonal, Or if there is the little corner 'boxy' thing that is drawn in some angles to indicate that it is a 90 degree angle). In truth, one cannot simply look at such an illustration and know that the angle is 90 degrees, because if it were in fact 89 1/2 degrees, or ninety-one degrees, you would not be able to tell from the drawing.

So here's the thing: First it is helpful to recall that all triangles on planet earth have a sum of 180 degrees of interior angles. That means no matter what the triangle looks like, when you take all three angles and add them up you will get 180.

Next, the question stated that two of the three angles add up to the third angle.

O.K., now, think of that equation, x +y = z as a balancing scale. (Which is what all equations are.)

And imagine you were to take the full quantity of the triangle's degrees and throw them across the room and hope to land them on the balance scale there, and, of course, have them balance perfectly. How do you think you would like to have your 180 degrees land?

All on one side of the scale?

No, that would not balance.

More likely, it would be great to see half the degrees land on one side of the scale and half land on the other. Half of 180 is 90, so it would be best it the 180 degrees get split into two equal piles in the air and have ninety land on each side.

Now look at the equation again:

X + y = z

The equal sign is the point of balance.

So you want 90 = 90.

Again x +y = z

90 = 90

So even though you may never be able to determine the degree measurement of angle x, or the degree measurement of angle y, you DO know what number they equal when added together (half of 180, or 90.)

Reread the question and see if you notice anything interesting (and easy!)...

Finally, just a day or two ago I wrote that I would explain how the triangle question we discussed here was not depicted well on our website. Fortunately, though we are still a little short-staffed this holiday season, an intern already stepped in and made the question and its diagram much better. So, if you haven't already, pull up the question* anew*. You will see the helpful change.

I hope this helps...

-- Mitch