## Spectre sighted in time for Halloween

Dave and Chaim. Composite: Chalkdust / Caroline Bonarde Ucci, CC BY 3.0

**Back in March, David Smith, Joseph Samuel Myers, Craig S Kaplan and Chaim Goodman-Strauss published the aperiodic monotile that they had discovered: the hat.**

Copies of this tile can fit together to cover the 2D plane in an aperiodic manner—the tiling never looks the same if is it shifted by any distance (except zero metres) in any direction. Importantly, this tile also cannot be tiled periodically.

Before the discovery of the hat, the best aperiodic tiles we knew about were the Penrose tiles—a pair of tiles that can only be tiled aperiodically.

There was, however, one issue with the hat: the aperiodic tiling that it made had to include reflections of the hat. This left an open question: does a tile exist that can only aperiodically tile the plane that doesn’t need reflections of itself?

In May, David, Joseph, Craig and Chaim answered this question with this aperiodic tile:

They call it ‘the spectre’ as it has no reflection, is 2½ hours long, and has a cameo by Judi Dench.

## Mathematicians win lottery, every time; brings happiness

**Boffins from the University of Manchester have shown you need to buy only 27 tickets to guarantee winning a prize on the UK national lottery.**

David Cushing and David Stewart say that among their 27 tickets, at least one will have two numbers in common with any of the possible 45 million different draws. Each ticket in the national lottery has six numbers between 1 and 59, and everyone who matches all six with those drawn shares the jackpot. Cushing and Stewart say we should settle for less—being happy to win *any* prize is good enough.

The Davids say there are no shortcuts: they’ve shown no set of 26 is enough. They set up the question in Prolog, a programming language that they describe as “temperamental, like a horse” to find the minimum number of tickets for different lotteries. But don’t go nuts—the smallest prize in the Lotto is a lucky dip ticket for the next draw. At £2 a ticket, you could easily end up winning less than you spent.

Since they published their findings, the Davids have been interviewed in newspapers, on the radio, and have even been the subject of YouTube videos. Their only disappointment? Nobody’s referred to them as “boffins” in print yet, and that they’re yet to win big on the lottery.

*Chalkdust* can help with one of these, at least!

## Not another one! Ninth Dedekind number found

**In April, mathematicians successfully determined the value of the ninth Dedekind number—32 years since the discovery of the last one.**

How did they crack this mathematical puzzle? And what exactly are the Dedekind numbers? This issue has the answers!