2017: New Year Doodles

For many, the days leading up to the dozen Big Ben bongs, which announce the New Year in my particular time-zone, are a time for fondly reflecting upon the highlights of the year we are about to bid farewell.

Sudden increase in UK budget deficit.

There appears to be a common understanding, however, that 2016 has been a year to forget. I know this because I am on the social media; the same social media whose predictions so let me down on the night of June 23rd. Cough - Brexit - Cough.

Matheminutes is very happy to pander to those Remainers/non-Trumpers/Fans-of-deceased-celebrities who have had quite enough of this year and are looking forward to the next.

So here goes.

There is a good reason that 2017 is already better than 2016:

Look, I memed.

Ignoring the fact that the last prime year (2011) saw severe earthquakes in Japan and New Zealand, riots on the streets of the UK, and the germination of a Middle-East power vacuum into which ISIS gainfully expanded, I can feel only excitement. It was, after all, the year in which matheminutes was born with an inaugural post about prime numbers.

We will have to wait a decade for the next prime year so let's enjoy this one while it lasts.

Why not begin with a silly puzzle?

Solved it? Don't expect me to give you the answer in this post. Maybe next time.

I fully expected that there wouldn't be a great deal to say about the number 2017. It does, however, emerge as the 63rd term in The Lazy Caterer's Sequence.

Imagine you are at a New Year's Eve party and have brought along a circular cake, pizza, pancake, Flammkuchen, Tesco Cinnamon Tear & Share bun, or any other delicious cylindrical, slice-able treat. Suppose it turns out that this is the most tediously boring party and you only have a knife with which to entertain yourself.

Why, oh why, isn't it midnight yet?

One legal possibility would be to find the maximum number of slices of cake/pizza/etc. you can produce with a certain number of straight cuts. One cut yields two slices. A second cut gives four. If you're careful with the positioning of the third cut, you can produce seven slices, and so on.

If you were both particularly bored and particularly skillful, you would find that 63 cuts would give you a maximum of 2017 slices. We should probably be more excited about this than about the fact that 2017 is prime. Lazy Caterer numbers are rarer than primes and the 64th cut will coincide with the year 2081. Having been born in 1981 I will, in all likelihood, be dead by then.

2017 is the only Lazy Catering Year I will experience during my life-time; if I invite you to a dinner party during the next twelve months, don't ask me if the bread rolls are home-made.

On the subject of sequences, I thought it might be fun (I have an unusual sense of recreational perspective) to produce a sequence beginning 2,0,1,7,...

We could use the fact that seven is one less than the cube of two and contrive to produce a recursive formula based on this:

This sequence proceeds 2,0,1,7,-7,8,335,-678,1190,37594185,... before blowing up faster than a [figurative comparison removed for sake of political correctness].

Alternatively, given that we do not need to involve digits 8 and 9, we could find a pattern modulo 8. The following diagram shows how such a sequence might begin:

It runs 2,0,1,7,0,6,7,5,6,4,5,3,4,2,3,1,2,0,1,7,... and has period 16.

Less geometrically pleasing would be a cubic sequence. We can force out a cubic that satisfies any first four digits. In this case, we could use the ordinal definition