## Combinatorial fairness property in division of goods

Given n agents, and m items where vi(g)≥0 is the value of item g for agent i, does there always exist a partition A1,…,An of the m items into n sets s.t. for all i,j∈{1,…,n}: ∑g∈Aivi(g)≥(∑g∈Ajvi(g))−(min Where if A_j is empty, the min is taken as 0. In essence this is a fairness property where … Read more

## Bound on number of steps needed for points to meet enclosing convex polygon

Let P be the set of equidistant points on the unit circle which are then randomly shuffled. They then take discrete steps towards the midpoint between the 2 points that they were originally adjacent to along a curve analogous to a pursuit curve. The illustration below shows a black line joining each point with its … Read more

## Numbers with a square sum arrangement

Informal version. For which $n>1$ can the numbers $1,\ldots, n^2$ be arranged in a square form such that the sums of the numbers in the little squares (consisting of $4$ numbers) are all equal? Formal version. For any positive integer $n$ we let $[n] := \{1,\ldots,n\}$. Let $n\in\mathbb{N}$ with $n>1$. If $\psi:[n]\times[n]\to [n^2]$ is a … Read more

## Social media for a mathematics related idea buckets

Are there any good social media platforms that can recommended for communicating ideas related to mathematics? The reason for asking is that I am in the situation that, albeit having studied math, I am not a professional mathematician and enjoy ruminating about mathematical problems from which I get the one or other insight I would … Read more

## Simplest examples of unique-solution and unsolvable-without-backtracking Sudoku-like problems

A The Sudoku game admits a broad generalization as follows : let r be an integer ≥2 and let X be a finite set, and X be a collection of r-subsets of X (i.e, a r-uniform hypergraph on X). We call any mapping X→{1,2,…,r} a coloring of X. Then, the Sudoku-like problem associated to any … Read more

## Iterated product of digits

It is well-known that the interated sum-of-digits function equally distributes the numbers from 1 to 10k−1 to the digits 1,…,9. And this holds true for any base b. For example, see the nearly decade-old MO question Sum of digits iterated. I want to ask a similar question for the product of digits. Let π(n) be … Read more

## Good quality data/packages for statistical/structure analysis of words in the English language

From time to time I find myself wishing to calculate basic statistics on words in the English language. For example, today I found myself wanting a graph of the number of English words vs. their length. Admittedly, such queries usually arise for me in the context of conversational/recreational purposes, but with the obvious links to … Read more

## Magic squares with specific properties

For what $n \geq 3$ does there exist an $n \times n$ matrix such that: All entries are in $(0, 1)$. Each row and column sums to $1$. Aside from the rows and columns, no other subsets of the entries sum to $1$. EDIT: I had a comment about $n = 3$ likely not being … Read more

## Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). This led to the following question. Let n be a positive integer. Consider the set C_n = \{1,\ldots, 2n^2\}\times\{0,1\}. … Read more

## Unusual matrix product associated with non-transitive dice

Not long ago, the Puzzle Corner of the magazine MIT Technology Review asked for a set of N dice that are non-transitive in the sense that there is a cyclic ordering on them, in which each die beats the next die in the cyclic order. I had not seen this particular question about non-transitive dice … Read more