A15. A Problem by Henry Dudeney

The houses on a long street have addresses 1, 2, 3, ... n. (Dudeney was British. Unlike the custom in most of the US, odd and even house numbers in Britain occur on the same side of the street. Assume all houses are on the same side of the street.) Call a house a half-way house if the sum of the numbers of the houses before it are equal to the sum of the numbers of the houses after it. Depending on n, there may or may not be a half-way house. For example, if n = 8, then there is a half-way house, namely 6, because 1 + 2 +3 + 4 + 5 = 7 + 8. You can check there is no half-way house if 1 < n < 8.

Find the value of n if you know that 50 < n < 500 and there is a half-way house.

You could do this easily by a brute-force search with a computer, so to make it interesting, no computer/calculator use is allowed.

I saw this problem presented online by the Mathologer. It connects with many fascinating parts of number theory, and there is an interesting connection with Ramanujan.

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.