It was to have been the Parmelan. But the weather forecast was not the best, albeit I kind of promised dry weather. So 6 of us showed up at 9.30 at the Divonne Lac parking, with Kobie. And Stephen met at 9.50 at Genolier, hoping for a gentle hike. I had hinted at 700m or so, with a gist of a plan to come down by train from St Cergue back to Sus Chatel.
Attending : Nathan, RichardS, Mark2ts, StephenL, BillW, Ross, and me PeterT, and Kobie. Apologies and excuses had been received from many, mostly plausible.
The forecast seemed to have worsened but almost 43% turned out in optimistic shorts.
We commenced by crossing the train track and heading up alongside the streamway where Kobie got some refreshment. After a slight break to take in the intimate stream landscape we continued on up, those at the front wettened by the close damp green leaves enfolding the path.
We soon arrived at Rob’s Gully, famous to the cognoscenti – which the team were rapidly becoming. At the top the path levelled out and a discussion ensued on prime numbers, in particular 17 which is a special number. And the fraction 1/17 has a coninually repeating 16 string decimal, a repetend.
153 is also an important number related to 17. Mark had some irrational pronouncements about rational numbers which none of us could make head or tail of. It all made for some light relief to the light rain which was beginning to fall.
No rest for the wicked and we continued up in what could have been Amazon rain forest. Muggy sticky and green.
The rain became a little heavier but it was not cold and we topped out at the Arizier Road in good order. A short stretch along the road, now partially in the cloud and driving rain, took us to the right turn which led through fields then into the upper forest. It was at this juncture in the lee of a large hedge, that the leader relented and offered the team a vote – to bail out and go directly to St Cergue for a pizza etc, as it was about midday already. We were saved by Mark’s phone which showed that the rain was about to end and that we could expect at least a two hour (relatively) dry spell. Vote cancelled!
What that meant was another 5 km and 300m ascent to the Fruitières de Nyon. The leader had a perception that the team were beginnng to become subdued, little was spoken as we trudged higher. A short revitalising stop was permitted before the final stagger up to the FdN under cloud and some wind, but no actual rain.
It was almost 1pm. We were alone and had several tables to spread out over, but huddled together for companionship.
At some stage black (or perhaps red) kites swirled over us in the maelstrom. A nesting pair of swallows were seen on the roof of the FdN itself.
A pleasant lunch, nobody choked so I (or they) did not need to use my LifeVac (I did provide a short demonstration just in case, as I had bought another soggy wrap from the Volg). Nathan helpfully suggested that I could reduce my choking risk by not choosing the soggy wraps. Mark offered strong black coffee and Nathan his Japanese whisky. Kobie had some good chews and part of a pork pie.
We were ready to go. The plan now was simply to descend to St Cergue the easist way, avoiding the intial steep and damp path. We took the road down.
We took the trail right past the youngsters on their zip wires at Basse Ruche and arrived in St Cergue at 2.38pm just after the train had left. Undaunted we tramped over to the boulangerie café (Le Ptit Gourmand) where Richard treated us all to a warm cuppa (each!).
Richard had not yet paid just as we realised we had 2 minutes before our 3.07pm train was due to leave. We held the door for him just as it was leaving and we settled down for the comfortable train ride back down the mountain.
From Sus Chatel it is only a short hop back to the cars.
Another memorable hike, which Kobie and I do quite regularly, but I found quite hard in the conditions. Both Kobie and I were exhausted that evening.
12 km. 740m or so of ascent.
Posting a Comment later from Mark:-
I can’t let an opportunity for a bit of maths pass so here’s what I think the guy was saying about 17 (already shared part of this with Peter). Also here’s a proof that any number with repeating digits must be rational.
- The decimal expansion of 1/17 repeats after 16 digits
- Perform the division by hand. Write out a few lines
0.0588
17) 1 00
85
150
136
140
136
4 - At each step there is a subtraction with the result (highlighted in red) must be less than 17. If the result is zero the decimal expansion terminates. If it is a repeat of a previous result the decimal expansion repeats from here. Therefore the repeat length can’t be greater than 16
- In general for any integer, n, the maximum repeat length must be n-1 and is usually shorter. I believe it must be shorter if n is not a prime since if n = r * s the maximum repeat length should be the product of the repeat lengths of r and s, i.e. (r – 1) * ( s – 1) < n – 1
- The repeat length is not always n-1 if n is a prime. c.f. 2, 3, 5, 11, also 13 (repeat = 6). So 17 is the smallest prime with this property.
- Perform the division by hand. Write out a few lines
- Any decimal with repeating digits must be a rational number (ratio of 2 integers)
- Assume X has repeating digits with a repeat length of n digits
- X may have some digits before the repeat starts, e.g. 1/6 = 0.1666…
Subtract off these digits and then multiply by powers of 10 until it is of the form
Y = ( X – X0 ) * 10m = 0.a1a2…ana0a1…an…
above is not strictly necessary but simplifies the next step - Now calculate Z = 10n Y – Y = a1a2…an, an integer
- Now work backwards
Y = Z / ( 10n -1 )
X = Y / 10m + X0 = Z / { 10m ( 10n -1) } + X0 , clearly a rational number - As an example try X = 3.142857142857…
X0 = 3, m = 0 , n = 6
Y = 0.142857142857…
Z = 106 Y – Y = 142857 an integer
Working backwards we get
Y = Z / 999999 = 0.142857142857… = 1 / 7
X = 3 1/7 = 22 / 7
Once again, thanks for a great hikeand for some mental stimulation., Cheers, Mark
