Dave Hewitt returns to his theme of multiple ascents with some number crunching and look at calendar rounds.
"All dates in time are visited" - Philip Larkin, Ambulances (slightly amended)
Last week I promised to bolt a few hard numerical facts on to the broad superstructure of musings about my 366 ascents of Ben Cleuch in the Ochils, so here goes.
I don't recall having thought about climbing Ben Cleuch on every day of the year before my first meeting with Alan Douglas, at the completion of his Ben Lomond calendar round in 1999. This was hardly surprising, as I was only "on" 165 Ben Cleuchs at that stage and still a long way short by any definition. But the notion interested and amused me, not least because I had studied statistics at university and knew that a substantial number of overall ascents was required to make a complete calendar spread likely. (It's effectively the reverse of one of the oldest statistical showpieces in the book - asking a class of 50 students what they think is the likelihood that at least two of them share a birthday, and then - after hearing various guesses based on an incorrect 366/50 calculation - revealing that it's 99% certain.)
I decided not to examine my own date-breakdown (which sounds like a plot device from Friends) until I was very close to 366 ascents total, so as not to skew the result any more than was inevitable. I'm a well-brought-up statistician, me. I did however know - because I'd asked and been told - that Alan Douglas's round of 366 different-date Ben Lomonds had been completed on his 578th ascent of the hill. I also knew that he had been deliberately targeting dates for some time, so it seemed reasonable to assume that 578 was below (or at least at the bottom end of) the expected number of ascents if the whole lengthy process happened to be random.
There had to be an equation for this - there's always an equation - so I decided to pick the brains of those more numerate than myself. A couple of hill companions are career mathematicians, and in June 2001 I asked one of them, Paul Prescott, if he knew how to calculate the probable number of "different" ascents for any given number of overall ascents. Paul's a clever chap, and even though we were picking our way down an alarming slope on Am Faochagach when I asked him, he simply said "Give me a minute," then went quiet. Eventually, after a bit more than a minute (I didn't actually hear his brain whirring), he said, "one minus one over e". And, as far as I'm aware, he was right.
Allow me a brief divergence into the land of probability. e is the exponential constant, the base of natural logarithms, an irrational number along the lines of pi, one that goes on for ever. The first few terms are oddly repetitive and easy to memorise: 2.718281828459045... and slotting 2.718 into Paul Prescott's Am Faochagach equation gives 1 minus 1 over 2.718, or 1 minus 0.3679, or 0.6321. How does this relate to a calendar round of hills? Well, one way to use the result is to turn it into a percentage - 63.21% - and then apply it to the overall group of "events". Hence 63.21% of 366 is 231.3486, so 231 is the likely number of different calendar dates for 366 ascents of the same hill assuming no other influencing factors - although even that is slightly off due to 29 February being only a "quarter" day. (Note that Alan Douglas's 366 Ben Lomond dates required 578 ascents, and that 366 is 63.32% of 578.)
Still with me? Good. The "no other influences" is a big deal, however. All sorts of things affect our lives and our leisure time, some plainly apparent, some less so. We tend to lapse into routines, such that certain times of year, or certain days of the week, become more or less likely for hill activity. The most familiar of these routines is that many people do the vast bulk of their walking at weekends - although that in itself need not greatly affect the overall distribution, as calendar dates cycle through the week due to the simple fact that neither 365 or 366 divides precisely by seven. It ought to be fairly obvious that if someone climbed a hill every Saturday without fail, then over time every date would be ticked off. (This process would take just over 10 years, again the leap day is a factor. For example, if someone had climbed Ben Nevis on 1 January 2000 - a Saturday - and continued doing so every Saturday thereafter, the calendar would be completed on 27 February 2010. Who knows? - someone could be busy with precisely this, even as I write...)
A much more influential negative factor is that people often aren't available to climb hills on certain regular dates, due to other commitments. Wedding anniversaries and the birthdays of family members are the most obvious examples of this. ("If you go up that damn hill today instead of treating me to lunch, then I'm filing for divorce...") Vicars/ministers/priests have difficulty with Sundays, imams with Fridays, and so on. But many of us also have at least one "block booking" at some regular point in the year, precisely the kind of thing that hinders target-bagging. My other mathematical hill-climbing friend, Ken Stewart, is a good example. Every year, at the start of November, he's the chief organiser for the Glasgow chess congress, so that weekend and the week preceding rarely provides any free time at all. (Plus it's the short-daylight time, so he isn't able to nip off for a quick bag of an evening as might be possible were the congress in May or June.)
Another way in which the statistical playing field tends not to be particularly flat is that the aspiring calendar-completer will already know the "status" of a few dates without checking. In my own case I was aware of having been up Ben Cleuch on the first seven days of April - during the initial Alva binge in 1997 mentioned last week. Christmas Day and Ne'erday had also been ticked off, deliberately targeted last year when pre-Christmas flitting made it a rare season in which neither Tessa nor I visited our respective sets of parents in northern England. I'm normally far more likely to climb Coniston Old Man or Crich Stand on Christmas Day than I am to climb Ben Cleuch, so I grabbed the chance while I could.
I was also pretty sure that I hadn't been up on either my birthday (10 July - stick it in your diary) or the bugbear of 29 February. But I knew that 11 September had been done, as my first knowledge of the Twin Towers situation came on returning to the car 20 minutes after the second plane had hit. (Whereupon, news-ghoul that I am, I raced home to watch the TV coverage.) Other than that, it was all a blank and a blur - I couldn't recall, for instance, any of my early dates until looking them up for last week's article.
Anyway, I'm waffling. Time for a few hard stats.
* The number of different dates for the 366 ascents was 236, pleasingly close to the theoretical figure of 231. I'd run this past the relative hills newsgroup (http://groups.yahoo.com/group/rhb/) by way of a competition, and the consensus seemed to be that the real figure would be lower than 231. The 13 guesses ranged from 190 to 293, both of which would be well into the tails of the distribution. The prize was shared by Iain Cameron and Alan Smeaton, who guessed 235 and 237 respectively. My own initial hunch back in 1999 - that the 366 ascents would include 210-215 different dates - shows why I've never risked a career in spread betting.
* If forced to guess beforehand which would be the quietest month over all 366 ascents, I'd have confidently said June and I'd have been right. June is when I'm most likely to be elsewhere, in the Hebrides given half a chance. So I've only been up Ben Cleuch 20 times in June, well short of the next lowest figure: 27 for each of November and December. Easily the highest is 43 for April: the joys-of-spring effect, no doubt.
* Also easy to guess - although near-impossible for the outside observer - was Thursday being my "busiest" day. For several years my regular copy deadline has been late Wednesday / early Thursday, and I've developed the habit of zipping off for a recuperative blast round the tops once the week's words have been shoved down the wires. So whereas the six non-Thursday days are tightly bunched (54 Wednesdays, 50 Mondays, 49 Fridays, 44 Saturdays and 40 for both Sunday and Tuesday), Thursday itself is well clear with 89 ascents.
* Combining these two analyses produces the odd discovery that I've never climbed Ben Cleuch on a Tuesday in June, the only blank day/month combination.
* As to the actual 236 different dates, April has the fewest gaps, with only three dates as yet unticked (8, 13, 23). June is the quietest, with 15 dates blank. Five months have a dozen blanks, two have 11 and one has 13.
* The 236 "done" dates mean 130 blanks, and the longest run is five, from 4 June to 8 June. The four four-blank runs are spread around the year: 29 January to 1 February, 4 to 7 March, 18 to 21 June and 18 to 21 October.
* The longest sequence of done dates is nine, three times: 30 March to 7 April, 14 to 22 April, and - less predictably - 6 to 14 December. The latter could be due to the Christmas shopping factor - evading the Christmas shopping, that is.
* I had no idea what would be the highest number of ascents for any one date, but it turned out to be five - on both 6 January and 18 July. The January example is odd, in that the ascents came in consecutive years, 1998 to date. Again, I had no notion of this until working it out - and I'm not sure whether knowing it will now make me more or less likely to climb Ben Cleuch again come 6 January 2003. Three dates have seen four ascents: 4 and 26 August and 22 December - that shop-avoidance ploy again.
* I reckon that 51 of the 366 ascents have been made in company, defined as setting off with at least one friend and reaching the top with them. That doesn't make 315 ascents strictly solo however, as I've periodically arranged to meet people on top - always good fun. It is however safe to say that at least 300 ascents have involved just me and my thoughts. All 32 July ascents have been made alone. Imagine that.
* Bivvymeister Ronald Turnbull will be appalled, but I must confess to never having spent a full night on Ben Cleuch. I keep meaning to rectify this - a couple of neat dossing sites spring to mind - but as yet it hasn't happened. There have however been over a dozen "moonwalks", ascents begun and ended in the hours of darkness, plus goodness knows how many descents in the dark due to sunset afterglow lingering.
* All three "centuries" have come at the back end of the year: the 100th ascent on 22 November 1997, the 200th on 19 December 1999 and the 300th on 19 September 2001. On both the 199th and 299th ascents I've been accompanied by my friend Richard Webb, up from Wolverhampton for the occasion, and we intend to keep this going for as long as we're both able. Already Richard has been warned to leave a day clear for 399 sometime in March or April next year.
Will I now try and complete the calendar, after the manner of Alan Douglas on Ben Lomond and Tom Bell on the selfsame Ben Cleuch (it took him over 1,000 ascents total)? Probably yes, although it could take a while as I'm notoriously dilatory about such things. Plus I'm likely to forget to keep tabs - eg it wasn't until returning from ascent 367 (12 September) that I noticed this was a "new" date. Nos.368, 369 and 370 were all repeats, and I'd be surprised to add 20 dates per year; indeed, I'm unlikely to seriously apply myself until past the 300-date mark.
There's also the diminishing-returns problem that as completion approaches, the slightest glitch could cause a substantial delay - especially if 29 February is missed when it comes round. So don't expect to hear of my having completed the calendar for the best part of a decade, if that. Certainly I'd be very surprised to dip under Alan Douglas's overall completion tally of 578 ascents, as this would require filling 130 gaps in 212 ascents (or 129 gaps in 208 as it now stands).
Whatever - it's just a game, and one that should keep me fit and content - which is what any kind of game or hill list or target ought to do. Lists and agendas, after all, are really just ways of structuring happiness.
Dave Hewitt
3/10/2002
Dave can be contacted at Dave.Hewitt@dial.pipex.com
