## 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 and put the elements of $Y$ into decreasing order $y_1>y_2>\cdots >y_n$

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

## Escaping elevens

From a Romanian Selection Test:

What is the greatest possible length of a sequence of consecutive positive integers, so that none of the integers have a digit sum divisible by 11?

## Children and Chocolate

Adapted from a Maths Battle in London:

Several children sit around a circular table, on which lies a collection of 100 chocolates. They proceed to take chocolates, after which it turns out that each child has taken either 6 fewer or 3 times as many chocolates as the child to their left. Prove that some chocolate was left on the table.

## Students

From the Tournament of the Towns:

In a school, more than 90% of the students study both English and French, and more than 90% of the students study English and German. Prove that more than 90% of the students who study both French and German also study English.

## 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?

## The Grid

A PuzzleCritic Original:

An empty 6-by-6 grid of squares is filled with numbers as follows. The first row contains the numbers {1,2,3,4,5,6} in some order. In each subsequent row, the kth number is equal to the position of the number k in the row above. For how many such grids are the first and last rows identical?

## 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?