Helen's Garden Renovation Project

It rained all morning, and now it is too soggy to mow the lawn, so here is the answer to the weed problem of 27 May 2007.

This is what the weed problem said:

A gardener has a patch of weeds. She pulls up 600 weeds every morning. Every afternoon the number of weeds goes up by 10%. After 10 days she has pulled up all the weeds (so she pulls up 6000 weeds altogether). How many were there to start with?

I received two solutions, both of which were different from my way of doing the problem, so you, lucky reader, get three solutions.

Solution 1: numerical
Sharon solved the problem by using Matlab code (Matlab is a specialist mathematical programming language) as follows:

D=zeros(10,1);
D(10)=600;
for n=9:-1:1, D(n)=D(n+1)/1.1+600; end

The solution is D(1). You can also do the same thing in a spreadsheet: put 600 in cell A1, then type “=A1/1.1+600” (without the quotes) into cell A2. Copy cell A2 to cells A3 to A10. Cell A10 then contains the answer, which is 4055.414, or 4055 to the nearest whole weed.

This solution would be accessible to the brightest GCSE maths students, as long as they realise that to increase something by 10% you multiply by 1.1, and therefore to undo the increase you divide by 1.1.

Solution 2: use of geometric progressions

For this solution you need knowledge of AS level maths (first year of sixth form). It is one of the harder geometric progressions problems. What you do is to write out the number of weeds at the end of each day, and spot the pattern.

Note that the number of weeds at the end of a day is given by 1.1(w-600) where w is the number of weeds at the end of the previous day. Also note that I am using the ^ sign to denote “to the power of” and * for “multiplied by”.

Suppose the initial number of weeds is n.
At the end of Day 1, there are 1.1(n-600) weeds which is 1.1n – 1.1 * 600
At the end of Day 2, there are 1.1(1.1n – 1.1 * 600 – 600) weeds, which is (1.1^2)n – 1.1^2 * 600 – 1.1 * 600
At the end of Day 3, there are 1.1((1.1^2)n – 1.1^2 * 600 – 1.1 * 600 – 600) weeds, which is (1.1^3)n – 1.1^3 * 600 – 1.1^2 * 600 – 1.1 * 600
At the end of Day 4, there are 1.1(1.1^3)n – 1.1^3 * 600 – 1.1^2 * 600 – 1.1 * 600 – 600) weeds, which is
(1.1^4)n – 1.1^4 * 600 – 1.1^3 * 600 1.1^2 * 600 – 1.1 * 600
Rearranging a little bit, this becomes
(1.1^4)n-600(1.1+1.1^2+1.1^3+1.1^4)

By now the astute AS level student will have recognised the last bracket as the sum of a geometric progression, and will replace it with the formula. This gives
Number of weeds at the end of Day 4 = (1.1^4)n – 600*1.1*(1.1^4-1)/(1.1-1)
which, believe it or not, simplifies to (1.1^4)n – 6600*1.1^4 + 6600.
By following the pattern, the number of weeds at the end of Day 10 is
(1.1^10)n – 6600*1.1^10 + 6600.
This, of course, is equal to 0, and therefore
n = 6600 – 6600/1.1^10
which also comes to 4055 to the nearest weed.

Solution 3: use of recurrence relations.
This solution was sent in by Dave. It requires undergraduate level maths. I reproduce his solution below.

I did the weeds problem by initially thinking of it as a continuous (rather than discrete) system i.e. the weeds are pulled up steadily at a rate of 600 per day (including all night) while simultaneously reproducing at an exponential rate of 10% per day. Although this is not quite the same problem (and therefore won’t have the same answer) it gives a clue as to the type of general solution that the discrete problem will have. The continuous problem can be expressed as:

Dw/dt = 0.1*w – 600 where w is number of weeds and t is time.

This has a solution of the form w = a*exp(0.1*t) + b where a and b are constants. The exact values of a and b can be found but are not important as it’s the type of solution that matters. For the discrete system, instead of a differential equation we have a recursive equation i.e.

A(n+1) = 1.1*(a(n) – 600)

In this the index n is equivalent to time t in the continuous system, so we can look for a solution of the form a(n) = c*d^n + f, where c, d and f are constants. Shoving this in the difference equation gives

C*d^(n+1) + f = 1.1*(c*d^n + f – 600)

Equating everything, d must be 1.1, and f = 1.1*(f – 600) giving f = 6600. So we have

A(n) = c*(1.1)^n + 6600

Also we know that a(9) = 600 = c*(1.1)^9 + 6600 giving c = -6000/(1.1)^9, so we end up with

A(n) = -6000*(1.1)^n/(1.1)^9 + 6600

The answer to the problem is the value of A(0), so substitute n=0 into the above equation and once again you get 4055 weeds.

So there you have it: three solutions to the weed problem. If you have got this far, I bet you really wish I’d mowed the lawn instead.

This morning, at about seven o’clock, just after my weekly forty-minute run, I looked despairingly at the weed in my lawn that does small yellow flowers and spotted a rather high-quality moth.

Thanks to http://www.butterflygarden.co.uk/moths.htm I can fairly confidently identify it as Tyria jacobaeae or Cinnabar moth. At least something likes the weed that does small yellow flowers.

The peony decided that procreation was more important than retribution, and did the following:

I spent a couple of hours pulling out more weeds and planting my tomatoes. I left it a bit late this year. One of the tomatoes keeled over just as I was about to plant it and broke its stem, but that still leaves me three plants, which is plenty.

The campanula portensclagiana has started making the most of having a pot to itself and is producing leaf and flower shoots with a fair amount of verve. I was worried that it might not be vigorous enough to hold its own in my garden, but I think it will do just fine.

I forgot to mention that on Wednesday I took some more cuttings from the lavender and rosemary at work. My first attempt, earlier this year, didn’t work at all. I suspect it is still too early for semi-ripe cuttings but I decided to try anyway. I also took some cuttings from the elaeagnus. So far they are not at all droopy or brown round the edges, but it is early days.

This morning was another bright and clear one, so I went out and did some more weeding. Here is a picture of me next to annuals growing in nice straight lines and hardly any weeds.

So far I have not thinned any of the annuals out, except by accident while weeding around them. I probably ought to have done that by now.

The new grass is now growing very strongly. Note the contrast between old lawn (mostly that weed that does small yellow flowers) and the new lawn (mostly grass). There are some gaps, especially at the border between the old lawn and the new, but I can fill those in when September comes. I did sow a little extra seed about two weeks ago because I was worried I hadn’t put in quite enough.

I had a spot of luck today. This morning, on my way to get a newspaper, I noticed that the skip in the drive of one of the houses I went past had several large plant pots in it. In the evening I went round to ask the owner if he would mind me liberating them from the skip, and he gave me permission, so I took them home. There are five 15 litre pots and two 12 litre pots, so they will be very useful to house my expanding plant collection.

Since it has been fine, I haven’t got around to posting the solution to the maths problem yet, but I promise I will soon.

It seems like a long time since I have been out in the garden, but it is only about a week. The May Bank Holiday weather was horrible – rain, cold and wind. I left my tomato plants out all night and I don’t think I should have done, because there is substantial leaf damage. However, the central shoots look fine, so I think they will survive. They are still in pots. I will plant them out very soon.

I saw a couple of not so nice sights in my garden. Firstly, there was a dead frog in my watering can. I don’t know whether it jumped in and couldn’t get out, poor thing. Then, after I’d gone indoors, I saw a magpie peck a smaller bird to death. I think it was a blackbird, because a couple of blackbirds jumped in and tried to chase the magpie away. It may have been their baby that was being attacked. In the end the magpie just picked up the bird and hopped away with it.

I pulled up no weeds today, but I did mow both the lawns. I also took the netting off the new grass and trimmed it with shears. I will post some photographs soon. At the moment it is not so much garden renovation I am doing as much as trying to keep the garden looking respectable while it tries to achieve the opposite.