From the Mailbag: Golfing Combinatorics

Sam’s dad is in a mathematical conundrum – so she’s asked Katie, one of our editors, if maths can save the day.

From the Sartorial Arts Journal, New York, 1901Dear The Aperiodical,

My dad is going away on a golfing holiday with seven of his friends and, since I know a little bit about mathematics, he’s asked me to help him work out the best way to arrange the teams for the week. I’ve tried to work out a solution, but can’t seem to find one that fits.

They’ll be playing 5 games during the week, on 5 different days, and they’d like to split the group of 8 people into two teams of four each day. The problem is, they’d each like to play with each of their friends roughly the same amount – so each golfer should be on the same team as each other golfer at least twice, but no more than three times.

Can you help me figure it out?

Sam Coates, Manchester

The Topological Tverberg Conjecture is False

Attention, Topological Combinatorialists! The topological Tverberg Conjecture, described as ‘a holy grail of topological combinatorics’, is false.

The conjecture says that any continuous map of a simplex of dimension $(r−1)(d+1)$ to $\mathbb{R}^d$ maps points from $r$ disjoint faces of the simplex to the same point in $\mathbb{R}^d$. In certain cases the conjecture has been proven true, but there have been found counterexamples in the case where $r$ is not a prime power, for sufficiently large values of $d$: the smallest counterexample found is for a map of the 100-dimensional simplex to $\mathbb{R}^{19}$, with $r=6$.

The result was recently unveiled at the Oberwolfach Maths Research Institute, which is situated in the Black Forest in Germany and regularly hosts bands of fiercely clever mathematicians. The disproof, by Florian Frick, is found in the paper Counterexamples to the Topological Tverberg Conjecture.

More Information

From Oberwolfach: The Topological Tverberg Conjecture is False, at Gil Kalai’s blog

Counterexamples to the Topological Tverberg Conjecture, by Florian Frick on the ArXiv

Florian Frick’s TU Berlin homepage

via Gil Kalai on Google+

Apiological: mathematical speculations about bees (Part 2: Estimating nest volumes)

This is part 2 of a three-part series of mathematical speculations about bees. Part 1 looked at honeycomb geometry.

Honeybees scout for nesting sites in tree cavities and other nooks and crannies, and need to know whether a chamber is large enough to contain all the honey necessary to feed their colony throughout the winter. A volume of less than 10 litres would mean starvation for the whole colony, whereas 45 litres gives a high chance of survival. How are tiny honeybees able to estimate the capacity of these large enclosed spaces, which can be very irregular and have multiple chambers?