Whenever I take part in an organised run or bike ride I think “never again”. Then I see another event advertised that I like the look of and start getting interested. There is a 100-mile charity bike ride taking place in London and Essex at the end of May this year. Frankly, I don’t think we’re ready and our new bike has been out of action for months (we ride a tandem). But, for next year, we could definitely be ready. Recently we joined a cycling group that promotes cycling near to where we live. Some of these people are very keen cyclists and I’m hoping for tips on the best way to train. We had a meeting with the group this week and were told that there are currently 228 members, an impressive number. I wondered where the number 228 would take me.

The City of Sheffield

The Crucible Theatre, Sheffield – Photo c/o unsplash.com

I considered whether any two places in the UK were exactly 228km (or 228 miles) away from each other. It turns out that the cities of London and Sheffield are exactly 228km apart according to this website.

I visit London at least once a year but have never been to Sheffield, even though it is much closer to where I live. It has a good university and I know it’s favoured by students who enjoy hill-climbing due to the nearby Peak District.

A friend and I have started meeting up for weekend-breaks around the country. We used to go to each other’s homes but have decided that meeting elsewhere is more relaxing and more fun. Last time we met up in Hereford. With my friend and these visits in mind I wondered if Sheffield would be a good place to meet. I love art and the theatre and Sheffield has both: the Crucible Theatre shown above is famous (here are all the theatres) and there’s also the Millennium Art Gallery. My friend enjoys visiting cathedrals (Sheffield has one) and there are several attractive-sounding green spaces. I’m interested in industrial history and Sheffield has a strong past in steel and silver – famous for knives and guns – and there’s a museum where you can learn more. I’m a bit of a foodie so I had a look at where to eat. I noticed that many restaurants offer a tasting menu – how wonderful – eight courses at one place. Top on my list would be places offering Sicilian food and Thai street food. All of this sounds amazing! So, yes, I’d definitely pay Sheffield a visit given the chance.

Refractable Numbers

A refactorable number an integer that is divisible by the number of its divisors

For example, 8 has 4 divisors (1 and 8, 2 and 4) and is divisible by 4.

An example of an integer which is not a refactorable number is 15 with 4 divisors (1, 15, 3, 5) but which is not divisible by 4.

The first few refactorable numbers are:

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296 and so on

There are infinitely many refactorable numbers.

What are the divisors of 228?

1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228

That is 12

And 228 is divisible by 12.

I’m not sure why these numbers are called refractable so this is a guess. ‘Refract’ means to change direction so perhaps refractable is another property of divisors that only some of the numbers possess: that is, looking at the divisor property from a different direction.

Last Thoughts

I love the way my meeting with my new cycling group has got me thinking about a visit to Sheffield with my friend and a property of numbers that I was unaware of.