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Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?


At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

Tree Tops

Age 14 to 16
Challenge Level
  • Which tree should you choose to minimise the cost of planting the forest?
Anh Minh from British Vietnamese International School Hanoi said:
In the first place European Larch is the cheapest tree to build when you start your farming because it cost the lowest.

  • Which tree should you choose to minimise your loss after 20 years?
Ziniu from Wantirna College in Australia and Anh Minh answered this question. Ziniu wrote:
After 10 years and 20 years, we need to thin the trees.
SS: +50k
EL: +55k
LP: +50k

The total cost for planting and thinning is:
SS: -70k
EL: -60k
LP: -80k

  • What would the profit be for each type of tree after 30, 40, and 50 years?
Anh Minh worked out which tree would be most profitable after each decade:
After 30 years, the Sitka Spruce will make the most profit
After 40 years, the Sitka Spruce will make the most profit
After 50 years, the European Larch will make the most profit
After 60 and 70 years, the European Larch will make the most profit.
After 80, 90, 100 years, the European Larch and the Sitka Spruce lost value, the Lodgepole Pine will make the most profit.

Best in 0,10,20,50,60,70: European Larch
Best in 30,40: Sitka Spruce
Best in over 80: Lodgepole Pine

  • What's the maximum profit you could make after 100 years?
Ben from Pembury in the UK began by looking at the final prices for selling the wood:
I saw that if you keep a pine tree for 90 years you get £1,476,000 which was the largest amount. Then you'd add [ £50,000] (money gained from thinning at 10 and 20 years) then you'd take away £130,000 (cost of planting) giving you [ £1,396,000] per hectare.

For the [other trees], using the above method you would select the highest amount then and [add the thinning price], then take away the [planting] price.

Ziniu found the profit per year to find a better strategy:
To find the optimal rates, we need to divide the amount of money gained by the years it takes to grow the trees to that stage so we can find the most efficient time/tree type to maximise profits. We need to also factor in the cost of planting and the gains of thinning as that may affect our results.

(amount gained from cutting + money gained from thinning - planting costs) divided by years it took to plant to that stage

Amount shown is in thousands of £

Tree name, after 30 years, after 40 years, etc
SS: 9.6, 11.1, 12.46, 12.73, 15.1, (no point calculating past this point as the value drops while years increase)
EL: 4.4, 10.23, 15.96, 18.73, 15.68, (no point calculating past this point as the value drops while years increase)
LP: 1.43, 7,15, 11,32, 14.5, 15.2, 15.39, 15.51 (no point calculating past this point as the value drops while years increase)

The most time and money efficient planting method is European Larch 60 years.

The question says to calculate the maximum amount of money earned in 100 years. 50, 20 and 10 are all multiples of 100. Planting and cutting trees at that time would maximise the efficiency of the plantation.

The most efficient plantation method would be EL 50 years, gaining 15960 £
a year and 1596000 £ after 100 years.

As well as replanting after chopping down the forest, Edward came up with a clever and detailed strategy involving planting different kinds of trees at the same time. Click here to watch Edward's video. Note that Edward has interpreted the thinning process in a different way.