## Archive for the ‘general math’ Category

It is possible to write many integers as the sum of the cubes of three integers. For example

99 = (-5)^3 + 2^3+ 6^3

A more complicated example is

91 = (-67134)^3 + (-65453)^3+(83538)^3

Your task is to find integers x,y and z such that

33 = x^3 + y^3 + z^3

Hint: This is not a trivial problem and will (probably) require the use of a computer. Extra credit given if you also post your source code.

**Update 3: **If you are serious about attempting to crack this problem (and some people seem to be, judging from the comments), the following reference may be of help. The bibliography includes other jumping off points for this problem.

- Michael Beck, Eric Pine, Wayne Tarrant and Kim Yarbrough Jensen:
**New integer representations as the sum of three cubes**Math. Comp.**76**(2007), 1683-1690 (Link)

**Update 2: **This problem is, in fact, a long-standing unsolved problem in number theory. If my memory serves me correctly, it is mentioned in the book Unsolved Problems in Number Theory

**Update 1:** 33 is very VERY difficult so why not use 16 as a warm up problem. MUCH smaller search space.

A friend recently asked me how to optimally tile the shape below on the plane such that no two instances touch.

Off the top off my head I suggested considering the minimal enclosing circle and then pack hexagonally but he thinks one could do better. I thought I’d ask on here to see what others might come up with.

I work in a beautiful old building at The University of Manchester called The Sackville Street Building. Yesterday, I took part in a very interesting historical tour of the building and was astonished at the amount of beautiful things that I had never noticed before and yet walk past every day.

For example, I was delighted to discover that one of the stained glass windows in the great hall was themed around mathematics (thanks to Samantha Bradey for the original image which I’ve heavily compressed for this blog post)

While on the tour I posted a quick snap of the above window to twitter using my mobile phone and received some nice feedback along with a few pictures of other mathematical windows. It turns out that Manchester’s Nick Higham has a close up of the above window for example. Also, Kit Yates tweeted about several mathematical windows in Caius College, Cambridge, one of which is below (showing images related to Venn and Fisher).

Do you know of any other examples of mathematics in stained glass? Feel free to contact me and tell me all about it.

My attention was recently drawn to a Google+ post by JerWei Zhang where he evaluates 2^3^4 in various packages and notes that they don’t always agree. For example, in MATLAB 2010a we have 2^3^4 = 4096 which is equivalent to putting (2^3)^4 whereas Mathematica 8 gives 2^3^4 = 2417851639229258349412352 which is the same as putting 2^(3^4). JerWei’s post gives many more examples including Excel, Python and Google and the result is always one of these two (although to varying degrees of precision).

What surprised me was the fact that they disagreed at all since I thought that the operator precendence rules were an agreed standard across all software packages. In this case I’d always use brackets since _I_ am not sure what the correct interpretation of 2^3^4 should be but I would have taken it for granted that there is a standard somewhere and that all of the big hitters in the numerical world would adhere to it.

Looks like I was wrong!

Consider a periodic sequence, seq_{p}, with period p

seq_{p} = *a*_{1}, *a*_{2}, …, *a*_{p}, *a*_{1}, *a*_{2}, …, *a*_{p}, *a*_{1}, *a*_{2}, …, *a*_{p}, …

Two sub-sequences, each of length N where 2< N< p, are produced by choosing two random start points in seq_{p} and using the next N-1 numbers. What is the probability that these two sub-sequences will overlap?

If you take M>2 such length N sub-sequences then what is the probability that **any** two will overlap?

For a more concrete example, assume that p = 2^19937 – 1 (yes, a very big number) and consider 10,000 sub-sequences each of length 10^9.

Disclaimer: I don’t have the solution to this and haven’t yet tried to find it

Consider this indefinite integral

Feed it to MATLAB’s symbolic toolbox:

int(1/sqrt(x*(2 - x))) ans = asin(x - 1)

Feed it to Mathematica 8.0.1:

Integrate[1/Sqrt[x (2 - x)], x] // InputForm (2*Sqrt[-2 + x]*Sqrt[x]*Log[Sqrt[-2 + x] + Sqrt[x]])/Sqrt[-((-2 + x)*x)]

Let x=1.2 in both results:

MATLAB's answer evaluates to 0.2014 Mathematica's answer evaluates to -1.36944 + 0.693147 I

Discuss!

Matt Tearle has produced a MATLAB version of my Interactive Slinky Thing which, in turn, was originally inspired by a post by Sol over at Playing with Mathematica. Matt adatped the code from some earlier work he did and you can click on the image below to get it. Thanks Matt!

Over at Playing with Mathematica, Sol Lederman has been looking at pretty parametric and polar plots. One of them really stood out for me, the one that Sol called ‘Slinky Thing’ which could be generated with the following Mathematica command.

ParametricPlot[{Cos[t] - Cos[80 t] Sin[t], 2 Sin[t] - Sin[80 t]}, {t, 0, 8}]

Out of curiosity I parametrised some of the terms and wrapped the whole thing in a Manipulate to see what I could see. I added 5 controllable parameters by turning Sol’s equations into

{Cos[e t] - Cos[f t] Sin[g t], 2 Sin[h t] - Sin[i t]}, {t, 0, 8}

Each parameter has its own slider (below). If you have Mathematica 8, or the free cdf player, installed then the image below will turn into an interactive applet which you can use to explore the parameter space of these equations.

Here are four of my favourites. If you come up with one that you particularly like then feel free to let me know what the parameters are in the comments.

I saw a great tweet from Marcus du Sautoy this morning who declared that today, June 28th, is a perfect day because both 6 and 28 are perfect numbers. This, combined with the fact that it is very sunny in Manchester right now put me in a great mood and I gave my colleauges a quick maths lesson to try and explain why I was so happy.

“It’s not a perfect year though is it?” declared one of my colleauges. Some people are never happy and she’s going to have to wait over 6000 years before her definition of a perfect day is fulfilled. The date of this truly perfect day? 28th June 8128.

**Update:** Someone just emailed me to say that 28th June is Tau Day too!

I do the administration for the Carnival of Mathematics and am very happy to announce that the 77th edition has been published over at Jost a Mon. If you are unsure what a Math carnival is then check out my introductory article or just read some past editions from either the Carnival of Math itself or its sister publication, Math Teachers at Play which is run by Denis of Let’s Play Math fame.

The next Carnival of Math is scheduled to be hosted over at JimWilder.com and the submission form for articles is open now. If you’d like to host a future carnival of math on your blog or website then please contact me for further details.