A little bit of magic

I first came across this problem years ago, somewhere online, but the exact source remains elusive:

Split \{1, 2, 3, ..., 2n\} randomly into two subsets X and Y, each containing n integers. Put the elements of X into increasing order x_1<x_2<\cdots <x_n and put the elements of Y into decreasing order y_1>y_2>\cdots >y_n

Prove that \lvert x_1-y_1\rvert +\lvert x_2-y_2\rvert + \cdots +\lvert x_n-y_n\rvert=n^2

Solution

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Five People

Here is a problem that appeared in a Maths Battle in London a couple of weeks ago:

Donald, Jack, Peter, Richard and Steven have, in some order, the surnames Donaldson, Jackson, Peterson, Richardson and Stevenson. Donald is 1 year older than Donaldson, Jack is 2 years older than Jackson, Peter is 3 years older than Peterson, and Richard is 4 years older than Richardson. Who out of Steven and Stevenson is older, and by how much?

Solution

Weights

From the Tournament of the Towns:

A balance and a set of metal weights are given, with no two the same. If any pair of these weights is placed in the left pan of the balance, then it is always possible to counterbalance them with one or several of the remaining weights placed in the right pan. What is the smallest possible number of weights in the set?

Solution