Answers to last issue's Brain Teasers

1. A solid, four-inch cube of wood is coated with blue paint on all six sides. Then the cube is cut into smaller one-inch cubes.  These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?
Ans: It is easiest to answer this question by looking at the figure that shows a big cube made up of 64 smaller cubes, 4 in each direction. The cubes at the corners numbered 1 will have three faces painted. There are 4 such corners at the top of the cube and 4 more at the bottom, so there are 8 cubes with three faces painted.
The cube marked 2 has two faces painted and there are 8 such on the top edges, 8 on the bottom edges and 8 on the vertical edges (the one shown). Hence there are 24 in all.
The cubes that have only one face visible are marked 3 in the figure and will therefore have only one face painted. On each of the 6 faces of the cube there are 4 such and so there are also 24 cubes that have only one face painted. What is left over is the total (64) minus those already counted, that is, 48-8-24-24=8 so there are 8 unpainted cubes that are the ones which are sitting as a 2x2 cube in the very heart of the big cube.
2. Four adventurers (Alex, Brook, Chris and Dusty) need to cross a river in a small canoe. The canoe can only carry 100kg.
Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and Dusty weighs 40 kg, and they have 20kg of supplies. How do they get across?
Ans: This is the usual optimisation problem. We need to send two men across so that one can row the boat back and we have to select the combinations so that we do not exceed 100 kg which is the maximum the canoe can carry.
Now Dusty cannot accompany either Brook or Alex since their total weight will exceed 100 kg. In fact, Alex cannot travel with any one at all, so he cannot be sent unless there is some other person on the other side who can row the canoe back.
So the obvious choice is that first Chris and Dusty row across; Dusty returns, since he is the lighter of the two. Now that Chris is on the other side, Alex rows across and Chris brings back the canoe.
At this point Alex is across and both Chris and Dusty are at the starting point. The two Chris and Dusty now row across and once more Dusty returns. You can see that Dusty's job is to essentially bring the canoe back. Now Brook rows across but since is is only 80 kg he takes the supplies along. As usual Chris brings the canoe back.
Finally, Chris and Dusty row across together for the final time and all of them are across.
3. How many steps are required to break an m x n bar of chocolate into 1 x 1 pieces?
You cannot break two or more pieces at once (so no cutting through stacks).
Ans: The simplest way to reach the answer is to understand that by breaking an existing piece horizontally or vertically, you merely increase the total number of pieces by one. For instance, if you break the chocolate once it becomes two pieces. If the chocolate was made up of two pieces in the first place, it is clear that one step breaks it up completely. Since the chocolate is made up of m x n pieces, to break it up completely you need m x n -1 steps.  Try it with a 4 x 2 piece of chocolate.
From www.mathisfun.com