E17. A 1-2-3 counting problem

The following problem seems at first to be quite difficult, but if you look at it the right way it isn't.

How many n-digit integers are there that contain no digits other than 1, 2, or 3, subject to the condition that any two consecutive digits differ by exactly 1.

This problem (for the n = 10 case) appeared in the ATMIM newsletter, Winter 2002, where it is credited to http://www.mathkangaroo.org, an interesting math enrichment and contest Web site.

I think this problem is too easy for me to post an answer, but if anyone asks for one, I will.

The buckling train track

Here's another gem from the ATMIM conference last month, suitable for any class where the students know the Pythagorean Theorem. Imagine a length of train track, two miles = 2 x 5280 ft long. To accommodate expansion of the track on hot days, the track is hinged at both ends and at the middle. If the track expands slightly, the middle of the track will rise, forming a shallow isosceles triangle. Supposed the track expands 2 feet. How high with the track be in the middle?

The teacher asks the students for guesses, which tend to be around 1 foot. Then he leads the students through the calculation of the answer. The height is the length of a leg of a right triangle where the hypotenuse is 5281 feet and the other leg is 5280 feet. This works out to be about 102.8 feet!