Sunday, 21 February 2021

Hospital tribulations

 This post is a departure from my usual style of blog entries because it is a personal record of a series of unfortunate events about my passage through the New Zealand public health system during my treatment for prostate cancer.

The story started with the surgery to remove my prostrate in January 2018 at the Dunedin hospital. When sewing me back together the surgeon stitched the tube that drains the wound into me so securely that it required a further operation under general anaesthetic to remove the drain. Well, these things sometimes happen and I would have thought no more about it if only the surgeon had had the grace offer his apologies for a clear error.

I seemed to recover well until, in late 2020, my PSA readings began to rise and a consultation was scheduled with my urologist in November 2020. This was a different person to the one who had conducted the surgery and I will suppress their names so as not to embarrass them. The consultation was by telephone and lasted about 10 minutes. In principle I have no objection to consultations by telephone but they have the unfortunate effect of not leaving the patient to make notes conveniently. I have since learnt that a transcript is prepared by the consultant which contains useful information for the patient. It is a matter of policy not to use email to forward it to the patient and I did not receive a letter by regular mail. Instead a letter is sent to the patient's GP.

The main conclusion from this consultation was that I should go to Christchurch for a PET scan and the surgeon undertook to notify Christchurch Pacific Radiology for them to schedule an appointment. PET scans, by the way, are not funded by the public health system and I was fortunate to have some private insurance for the $3000 fee. I was told that I should hear something soon.

Nothing then happened for several weeks and in the second week of January I phoned the Urology department at Dunedin hospital. It was clear from the reaction that the consultation had not notified the radiologists and he had now gone on leave. A flurry of activity by a very competent administrator resulted in a PET scan appointment for me on 21 January.

Since the appointment was at 11.00am my wife and I rose at 4.15am that day to drive to Christchurch. It was a smooth journey until we reached Timaru some 250kms from Dunedin. Then we received a telephone call from Pacific Radiology to cancel the appointment as the radio-active pellet required had not been loaded onto the plane. This was a somewhat low psychological moment and we had to return to Dunedin with another appointment arranged for the following week.

The PET scan happened on 28 January. I was told that the results would be sent to my consultant and GP within a day or two, and I asked for a copy for myself.

Two weeks then passed and I heard nothing so made another phone call to the Urology department. They had received the results but would not give me any details. I was aware of a level of embarrassment when I said I had heard nothing from them - and they told me I would see the consultant on 8 March, and they would write to confirm (email confirmation again being impossible but, mirabile dictu, my GP would receive a letter).

A further week passed and still no letter so I made another phone call. More embarrassment and I was told the letter would be sent immediately.

The very next day I received a phone call to say that the consultant would actually be on leave on 8 March but they could offer me a phone consultation on 24 February. I agreed to this with some misgivings and after receiving assurances that all would be confirmed by letter. Possibly I was being alarmist but I had absolutely no idea about the seriousness of my condition and I was uneasy about having to react over the phone to some possibly challenging news.

The next day I received the confirmation that my 8 March consultation was arranged (the one that had been cancelled the previous day) but I recognised that this had most likely been sent before the cancellation.

I write this on 21 February and will update the saga as it continues to develop.

24 February: I waited patiently by the phone from 30 minutes before the appointment time of 1.40pm. No call. After an hour I called the Urology department to ask what was going on - and received the message that this was outside their business hours (mid-afternoon). Then I texted the Urology department, received no reply, and one hour and twenty minutes after the appointment time, still not having heard, had to leave. At 5.40pm the consultant called. The PET scan had been inconclusive and he recommended to just monitor the PSA levels and that he would write to my GP about the next PSA test.

Saturday, 16 January 2021

Early memories of Peter Neumann

My friend and mentor Peter M. Neumann died on 18 December 2020. There have already been many tributes to him and fond memories recalled. Some of these are reported through the Queen's college memorial page to him. Here I recall some personal early memories that are maybe not so widely known.

I went up to Queen’s in October 1964 at the age of 18 and immediately met the 23 year old Peter Neumann. He became my mathematical mentor first as undergraduate tutor and then doctoral supervisor, and was easily the biggest influence on my development as a mathematician.

In the acknowledgements in my D. Phil. thesis (1970) I wrote “My chief debt is to my supervisor Dr. P. M. Neumann, whose interest and encouragement were unfailing. It is a pleasure to thank him for all his advice and to record my appreciation of his friendship.” I was to know this warm, courteous, witty and clever man for a further 50 years and we had many personal and mathematical interactions. Here I’d like to mention some earlier memories as his student which, looking back, were particularly formative for me..


It was common in the 1950’s and 1960’s for boys to be called by their surname so I was taken aback when Peter immediately addressed me as ‘Mike’ and invited me to call him ‘Peter’. It was also a surprise that he always greeted me (and my fellow tutees) warmly when we met going about Queen’s. But for all this familiarity he had high expectations of his students and spared no effort in his encouragement.


In my first long vacation he got all of us to write an essay on a topic chosen from a list of mathematical topics well beyond the standard curriculum. I remember one of these was ‘The art of M. C. Escher’ but the one I chose was on Hilbert’s problems. Peter gave me the background. David Hilbert, in a famous address to the 1900 International Congress of Mathematicians, had proposed 24 problems that, in his opinion, were the greatest mathematical challenges of the day. Some of these had been solved, some had faded into oblivion, and some were open. My essay was supposed to summarise the present status of these problems. Peter appreciated  that my mathematical knowledge would be inadequate to even understand some of these problems and offered help on request. I did indeed require help and the Oxford-Leeds correspondence that ensued gave me my first insights into the research mathematical literature and such tools as Mathematical Reviews. It also impressed on me how widely knowledgable Peter was.


I became increasingly aware of the formative role that Peter’s parents, Bernhard and Hanna, had played in Peter’s early life. He once told me, as a very young child, he had watched Bernhard prepare breakfast for the family, counting out the slices of toast in the mysterious sequence 1, 4, 9, 16, 25, …. He realised eventually that the rule must be that the differences increased as 3, 5, 7, 9, …. and was delighted when Bernhard revealed the sequence of squares rule.


While his parents were obviously key influences in Peter’s early mathematical life he became an independent mathematical thinker well before I met him, publishing his first single-authored paper while still an undergraduate. However he never forgot his debt to his parents and always spoke fondly of them. In 1969 he dedicated his splendid paper on BFC groups ‘to my father on his 60th birthday, with love’.


By the time I became his graduate student Peter had become the most versatile of a generation of young Oxford researchers in algebra. But he never talked down to his students or his colleagues and was willing to engage with them almost on demand. I remember in about 1969, when I had learnt the rudiments of the theory of group characters, hyperbolically proclaiming to him that this must be the neatest little topic in the whole of mathematics. I had some reason to hope that he might agree with this proposition since I had heard him lecture on William Burnside, one of the originators of character theory. It would have been easy for Peter to prick this pompous little bubble but, after some some careful thought, he offered an alternative opinion that the theory of complex variables was even neater. Since this theory is not even part of algebra it made me realise that taking an interest in areas outside my own would be a good habit to cultivate and that has stood me in good stead ever since.


I knew Peter in another role too. He and I were both members of the University folk-dancing society. Peter’s contribution to the dancing was in providing music on his violin. I do not know very much about Peter’s other musical activities but he was a superb asset to our folk-dancing many times being the only accompanist and (so it seemed) effortlessly sight-reading whatever we asked of him. Dancing and directing dancing has been an occasional activity of mine throughout my life and, on those occasions, I always think of Peter. His face would display a mixture of concentration and enjoyment, a combination which accompanied also so many of his mathematical activities.

Friday, 22 May 2020

Joining points and sums of squares

This posting is an extended comment on another entry in R J Lipton's blog Gödel's lost letter and P=NP. Lipton's post is entitled A Math Gift for All and is a clever but simple solution to a geometric problem wrapped in a deceptive narrative. You might enjoy reading the post before proceeding here although that is not necessary provided you accept my spoiling the joke. My commentary is on both the solution and the numerical deception.

First the problem (which came from Shmuel Weinberger's  book) and Lipton's solution. You are given 2n points in the plane no three of which are collinear, n of which are coloured red and n are coloured blue. Do there exist n straight line segments each joining a unique red point to a unique blue point (as in a matching) in which no two line segments cross?

Lipton teases us with this (see below) but to cut to the chase the answer is Yes by the following argument. Among all the ways of joining the red points to the blue points in a matching consider one where the total length of the n line segments is the smallest. In this matching no two line segments cross because if two lines R1B1 and R2B2 cross then R1, R2, B1, B2 are adjacent points in a quadrilateral and if we replace the diagonal crossing of R1B1 and R2B2 by the line segments R1B2 and R2B1 we will have another matching with smaller total length of segments.

I showed the problem to my son James, a high school math teacher, who found another argument. Begin by choosing a point P not one of the given 2n points but within the cloud of such points and consider any directed straight line L through P. Because of the choice of P not all the 2n points lie on side of L. For any such line L let RL and BL denote the number of red points on the left of P and number of blue points on the left of P. If these numbers are equal then we have divided the set of 2n points into two sets - one set on the left, one set on the right - each containing equal numbers of red and blue points; then we can solve the problem on either side and get a solution to the original problem. But if RL and BL are not equal we can rotate L about P. As we do so RL and BL change by 1 every time L passes past one of the given points and after 180 degrees of rotation the values of RL and BL will have exchanged; so at some stage we shall have RL = BL and can proceed as before.

Both solutions can be made constructive but James' solution seems to lead to a more efficient algorithm (although it may well depend on how much ingenuity you are prepared to use).

The second topic I want to discuss derives from the trick Lipton played on his readers. He introduced a constant c whose value was 102+112+122-132-142, drew the readers in for a few lines and then pointed out that c=0. This identity between consecutive squares is the second in a sequence of identities the first of which is 32+42=52. There is a sequence of similar identities in which k+1 consecutive squares add up to the sum of the following k squares; the next identity (k=3) is 212+222+232+242=252+262+272. In general the middle square of the kth identity is the square of 2k(k+1) which is readily proved by elementary algebra.

Wednesday, 20 May 2020

Beetles around a circle

I recently came across a lovely combinatorial problem in R J Lipton's blog Gödel's lost letter and P=NP. You can read about it in that blog, together with its provenance, and its beautiful solution. However, in order for my comments on it to make any sense I need to recap the problem and its solution.

9 beetles are placed around a circle and the distances between them are the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23. They start walking around the circle at the speed of 1 unit of distance per second but their directions are chosen randomly. When two beetles collide they reverse their directions but maintain their same speed. After 50 seconds and many collisions what are the distances between the beetles?

The remarkable answer is that the set of distances is still that same set of prime numbers.

The red herring in the problem is that the presence of prime numbers is irrelevant: no matter what the initial distances are the set of distances returns to that initial set after 50 seconds. The only thing that matters is that the sum of the distances is 100 which is therefore the length of the circle's circumference. The key insight is to imagine that each beetle carries a flag and whenever two beetles collide they exchange flags. Thus each flag continues to travel in the same direction it began with and so, after 50 seconds, it has travelled 50 units of distance and therefore has travelled halfway around the circle to the antipodal point. Since the final position of the flags is a reflection in a centre point mirror of the initial position, the set of distances between the flags is is the same as the original set of distances. However every flag has a beetle on it and therefore this is also true of the set of distances between the beetles.

Of course after 100 secs every flag has returned to its starting position and again the original distances are restored. However it is hardly ever the case that the beetles themselves have returned to their original positions. My friend Dennis McCaughan asked "When do the beetles get back to their original position?" and he and I quickly worked out the answer.

It is obvious that the relative order of the beetles around the circle never changes (as two beetles never swap places). After 100 seconds each beetle has moved to one of the original beetle positions, say through t original clockwise beetle positions, and t is the same for every beetle; t can be positive or negative. The clockwise distance a beetle has travelled is its clockwise drift, a sum of t original consecutive distances.  The total clockwise drift is the sum of the individual drifts and in this sum each original distance occurs t times. So the total clockwise drift is 100t. 

Initially let there be x beetles moving clockwise and y=9-x beetles moving anticlockwise. Since every collision is between a clockwise beetle and an anticlockwise beetle (turning a clockwise beetle into an anticlockwise one and vice versa) x and y do not change. We can think of d=x-y as the speed of clockwise "drift", the amount of clockwise distance consumed per second. Therefore the total clockwise drift after 100 seconds must be 100d. Hence 100d = 100t and so d = t.

We can now determine when the beetles will all return to their original positions. If d=9 or -9 then, after 100 secs, the beetles will be back in place after 100 seconds since each beetle will have moved through 9 original beetle positions. If d=3  or -3 then after 100 seconds each beetle will have moved through 3 original beetle positions clockwise or anticlockwise. So 300 seconds brings them back to place. For other values of d, a full 900 seconds is required.

Sunday, 17 May 2020

A puzzle about disease transmission

I heard this puzzle over 30 years ago when it was circling in North American Computer Science academic circles. The current pandemic and how to stem its spread reminded me of the puzzle. It concerns three sailors, one prostitute and two condoms. Stop reading now if you find that scenario too unsavoury.

The four characters have agreed that each sailor will have intercourse with the prostitute and the problem is to do this safely with only the two condoms at their disposal. What does "safely" mean here? Certainly they wish to guard against the sexual acts resulting in conception but they are all aware that, given the company they have been keeping, one or more of them may have one or more STDs and they don't want any further infection to be transmitted. There is a short discussion of the problem in this Reddit thread.

The solution is not that hard although it is quite easy to find convincing "almost" solutions. Here is one. Sailor Donald wears the first condom and has his wicked way. Then sailor Eric puts on the second condom and then the first condom over it and does the deed. Finally sailor Jared turns the second condom inside out, puts it on, then puts on the first condom over it and completes the action.

This fails in one respect only: Donald has infected the inner surface of the first condom; during the second act this infection is transmitted to the outer surface of the second condom, and so Jared can become infected by Donald's poison. A further incorrect solution appears in the Reddit thread (and then corrected).

The true solution is very similar. Donald uses both condoms (the second inside the first); Eric uses the first condom, Jared turns the second condom inside out and then dons the first condom.

In each sexual act the prostitute is only in contact with the outer surface of the first condom and so she cannot contract a fresh infection. Each man's member touches a condom surface untainted by any infection. All quite easy and possibly a puzzle and solution known to many.

I offer one final twist. Suppose there are 2n+1 sailors and n+1 condoms (but still one prostitute). How about safe sex now?

Let's number the sailors 0 to 2n and the condoms 0 to n.
For each i = 0 to n-1 sailor i wears condom i and also condom 0 over it
Sailor n just uses the single condom 0
For each i = 1 to n sailor n+i wears condom i inside out and also condom 0 over it.


Monday, 31 December 2018

Widening the franchise

Before the second world war the voting age in most countries was 21 but in the second half of the twentieth century the age of 18 gradually became very common. For example, the USA passed the 26th amendment to its constitution in 1971 largely because it was felt that to be conscripted at age 18 and not allowed to vote was anomalous. There are a few countries where the voting age is 16 (such as Austria) and a very few where it is higher than 18 (such as Kuwait and Bahrain). In many countries there have been movements to bring the age down to 16 (for example, the UK, Australia, and New Zealand) but these attempts were abandoned (although, for the 2014 Scottish Independence referendum, the voting age was 16).

The reason that 18 has emerged as the most common voting age seems to be that 18 is seen as the age when children become adults. That this transition should occur instantly on an eighteenth birthday is obviously a fiction and therefore it is legitimate to ask whether we have got the age right, and to ask whether adulthood is the criterion we should be using.

The assumption that adulthood is a prerequisite to vote is very ingrained. Surely, it is said, a certain maturity is essential before a considered vote can be cast. And yet the right candidate to vote for and the right set of policies to support are generally not decisions that are arrived at by the tools of knowledge and logic; because, if they were, there would be no dispute at the ballot box. Once one accepts that opinions are driven by prejudice and emotion who is to say that a 16 year old is not as qualified as an 18 year old? Let's be honest and admit that most people vote out of self interest even if they are able to rationalise their decisions.

So why should the voting age not be 16? But isn't that as arbitrary as 18? If you are thinking along those lines consider the very bold proposal by Cambridge Professor David Runciman. He suggests the voting age should be lowered to 6! He has admitted that his proposal is made with a certain tongue in cheek but is ready to defend it anyway. For example, what of the objection that children will just vote as their parents tell them? Why should they? Women did not vote the way their husbands said they should when they got the franchise?

For me, the most compelling reason for taking Runciman seriously is that our electorate is already top heavy with old people. The most compelling issues of our age - climate change, whether Scotland should have independence, how should institutionalised racialism be addressed in the USA and other countries - are long term issues. Why should old people have any say in these? Leave it to the citizens who have the most skin in the game. After all if, like me, you are appalled at how the majority of the UK voters voted in the EU referendum, consider what might have happened if the franchise had consisted also of 16 and 17 year olds (a proposal that was seriously floated in 2015). Don't you think their youthful good sense would have prevailed over the grey vote which, on the day, carried the Leave side over the line?

Sunday, 29 April 2018

The trauma of the United States

Recently I read "Loaded" by Roxanne Dunbar-Ortiz. It is a history about the US Constitution's Second Amendment and the mythology surrounding it. The Second Amendment is once again in the news for a reason that occurs with depressing frequency. This time it is the shooting that occurred at the Douglas High School in Florida in February and which has resulted in a mass movement Never Again led by some of the survivors at the school.

I must confess that I began Dunbar-Ortiz's book with pessimism. I expected to read the usual arguments for gun control and the usual counter-arguments and I thought cynically that believing anything would really change was just pie in the sky. That is not exactly how it worked out.

The first thing to say is that the book has a deep and well-researched thesis that the history of the USA from even before its foundation contains a number of deeply disturbing causes for the national gun culture that go way beyond the usual excuses. This is not about personal freedoms endowed to the citizenry by the Founding Fathers. It is not about the national hunting culture. Nor is it about the distrust fostered by the NRA that the government wants to disarm the people to enable their suppression ("from my cold dead hands").

No, it is about slavery and genocide. The US Constitution is not about all men being created equal; it is about the privileges of white male landowners and the second amendment is why white male landowners needed guns. They needed guns to contain their slaves through patrols looking for escapees. And they needed guns to carry out the systemic slaughter of Native Americans so that their lands could be seized. It is interesting to reflect that a key reason for the Declaration of Independence is that King George III tried to restrict the colonists' "right" to seize territory west of the Appalachian mountains whose inhabitants were described as "merciless Indian Savages".

This is not an interpretation of their history that most Americans will like. They are more used to a narrative in which heroic settlers fought for their freedom from an oppressive colonial tyrant and then, through the nineteenth century, expanded ever westwards claiming land for themselves. A history in which brave cowboys and fearless rangers protected farmers from violent attacks by Indians and they came into their manifest destiny much like the Old Testament Jews settling Canaan (just after God gave them their marching orders to kill every one of the original inhabitants). These myths protect them from their murderous pasts and are part of the defence of their wide gun ownership. They revere the part that guns played in taming their land, they laud the bravery of the family man whose gun is to protect his nearest and dearest as part of a long tradition, and they hold their constitution in almost superstitious awe.

So, on reaching the end of the book, I had a sense of hopelessness. How on earth could one penetrate these myths so that a rational discussion about how to go forward could be held? After a few days I began to read some other historical material in the same vein and this left me feeling a little more optimistic. To begin with the USA is not the only country burdened by a very shameful history. The Doctrine of Discovery promulgated by Pope Alexander VI in 1493 initially gave the Spanish permission to take possession of colonise any lands they  discovered which were not under the control of a Christian ruler and it became the justification for later European powers to colonise at will (and the American colonists inherited this idea from the British). The world is still suffering from the aftermath of colonisation by force but other colonial powers have at least begun to recognise their catastrophic agency with apologies, reparations, and deliberate reconciliation. The American haven't really started down this road but they did come late to the game - maybe their eyes will be opened in due time.

But also we should not overlook that there are movements in the USA that are trying to confront their racist misogynistic culture: the Woman's Movement, Metoo, the LGBT movement, Black Lives Matter and more. Some of these movements have extraordinary charismatic leaders who recognise the extreme difficulty rank and file Americans have in facing up to their past. I encourage you to view the lecture by Mark Charles; he is a native American who, for all his criticism of the oppression of his people, ends his lecture with some prescriptions that might change the discourse. We have a long way to go (and other colonial powers are still travelling that road) but it is important to try to keep reason, tolerance and understanding alive - and maybe in a couple of hundred years we can emerge in an enlightened sunshine.