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E33. Tiling a checkerboard

This problem was shown to me by my student, Apratim Roy. Though it involves only elementary concepts, I found it rather difficult, and thought the solution was very surprising and elegant.

You are given a standard 8 x 8 checkerboard, with one square removed, and 21 3 x 1 tiles. In other words, there are exactly enough tiles to cover the modified board. Your task is to find a way to do this, without cutting any tile.

(a) Find out what square must be removed for the task to be possible. (4 possible answers).
(b) Describe the tiling. (Many possible answers).

You might guess that the removed square needs to be one of the four corners of the checkerboard, but you would be wrong.

A13. Points of tangency of an ellipse and a circle

Let E be the ellipse with equation x2/4 + y2 = 1 and C(r) be the circle with center (1,0) and radius r. For which values of r do the curves E and C(r) have point(s) of tangency?

This is fairly routine, but still a bit challenging to find all solutions.

A12. Fifth powers final digit - generalized.

If numbers are expressed in base b, for which b is it true that n5 and n end in the same digit for all positive integers n?
This is an obvious generalization of Problem E32.

E32. Fifth powers final digit

The following rather neat problem occurs in Challenging Problems in Algebra by Alfred S. Posamentier and Charles T. Salkind. I think it is suitable for a bright high school student or, with some hints, even for average high school students.

Prove that n and n5 always end in the same digit (in ordinary base-10 representation).