201. A Cubic Angle 2012-1-20

Imagine the following picture, draw it, or model it with a Rubik's Cube^TM. You see three faces of a cube: the top, the front, and the right side. Point A is at the back (far), left corner of the top face. Point B is at the right, front (near) corner of the top face. Point C is at the bottom, left corner of the front (near) face.

Line segments AB and BC form an angle at vertex B. What is the measure of that angle in degrees?

Answer

Turn the cube, so you can see corners A, B, and C, surrounding the unique corner that is at the other end of an edge from each of them.

Clearly, A is a diagonal away from B, which is a diagonal away from C, which is a diagonal away from A.

Triangle ABC is an equilateral triangle, so angle ABC is 60 degrees (180 / 3).

If the length of the side of the cube is s, then the length of a diagonal is s √2. The line segments, AB, BC, and CA are all diagonals, so they have the same length, so the triangle is equiliteral, so it is equiangular, so angle ABC is 60 degrees.  

202. Are you smarter than an eighth grader?

Here's the question that the champion answered at a recent middle school math competition:

Question

A bag of coins contains only pennies, nickels and dimes with at least five of each. How many different combined values are possible if five coins are selected at random?

Answer:

21 unique values: 5¢, 9¢, etc. If you know "the trick" for this sort of problem, you can do this in your head in less than 15 seconds. (It took me 15 minutes to realize that I could solve the problem in 15 seconds!)

The champion was the 2012 Raytheon MathCounts National Competition champion.