Answers to last issue's Brain Teasers

1. A puzzle to puzzle you

	You are the only survivor of a plane crash. You are standing on flat ground. There is nothing around you: no trees, no houses, nothing at all. You try to find out where you are. You leave your start walking and go one km south. You don't see anything at all. So you go one km west. Still nothing. So you go one km north and find you are back to where you started, at the plane crash. How is this possible?
	A variation on this puzzle is that you are again lost. This time you go one km north and then one km south, but you are not where you started. What is going on? How can this be?

Answer:

	Whenever we think of directions, we think of Earth as being flat. This is typically true, since the Earth is so large. However, we live on the surface of a sphere (almost spherical). The latitudes and longitudes that tell distances on Earth are perpendicular to each other only near the Equator.
	So there are places on Earth where longitudes are not parallel such as at the poles (the first version of the puzzle depends on this).
	This is more easily understood with the help of a figure. In the first case, the person clearly crashed at the North Pole. He walked south along one longitude, as shown, then walked west until he reached another longitude 1 km away, then simply walked back north to the north pole.
	In the second case, he must have started at the point marked "2" on the figure. In he was on the Equator, for instance, he would have been very far from /near to his starting point, as can be seen from the dotted path starting on the Equator in the figure.

<earth.gif> or <earth.eps>

2. Numbers game

	Think of a 10 digit number that is divisible by 9. Reverse the order of the digits. Is it still divisible by 9?
	Think of a 10 digit number that is divisible by 11. Reverse the order of the digits. Is it still divisible by 11?

Answer:

First, some facts:

1. A number is divisible by 9 if and only if the sum of its digits (in decimal notation) is divisible by 9. So 45 is divisible by 9 since 4+5=9 and 123456789 is also divisible by 9! 
	Since the sum of the numbers is the same whether you count backwards or forwards, if a number is divisible by 9, so is its reverse. 
	The property of addition giving the same answer forwards and backwards is called commutativity.

2. To decide if a number N is divisible by 11, let the digits of the number in decimal notation, from right to left, be ABC...K. Add up the digits in the odd positions: A+C+E+... to get the number So. Add up the digits in even positions: B+D+F+... to get Se. Now N is divisible by 11 if and only if the difference So-Se is divisible by 11. It doesn't matter if So is smaller or larger than Se; always consider the positive difference. This is called the modulus by mathematicians.
	Reversing the order of the digits does not change the quantity So-Se defined above. So if N is divisible by 11, so is its reverse.