You and I play a game involving successive throws of a fair coin.
Suppose I pick HH and you pick TH. The coin is thrown repeatedly
until we see either two heads in a row (I win) or a tail followed
by a head (you win). What is the probability that you win?
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
Which of these games would you play to give yourself the best possible chance of winning a prize?
No one completed the solution to this problem so we will put it
into Tough Nuts for you to return to when you have time. There were
several attempts and some of you were able to identify that there
are 784 possible arrangements of three factors of a million.
The first step was to identify that 1000 000 = 10 6 =
2 6 x 5 6 and then to consider this as the
product of three factors i.e.
However there are repetitions here because 2 3 5
3 x 2 2 5 2 x 2 1 5
1 is the product of the same three factors as 2
2 5 2 x 2 3 5 3 x 2
1 5 1 .
So there is still some work to do! Good luck.
Giles Cooper & Mike Hood thinks there are 139 such factors,
he has produced a list of factors for example
1 x 1 x 1000000
1 x 2 x 500000
1 x 4 x 250000
80 x 100 x 125
100 x 100 x 100
This is correct, however I also liked the attempt by Mike Hood,
who tried to use the idea of combinations of factors. This idea
seemed to be a ``less exhaustive'' approach and begins to give a
better insight into what is going on. His solution uses the fact
that $1,000,000 = 2^6 x 5^6$. I think this is worth pursuing. Think
about the number of ways you can combine the six 2`s and the six
5`s in the three factors. You then only have to consider the number
of unique combinations of each of the two sets of arrangements. For
example - you could choose $2^6$, $2^0$, $2^0$ and $5^6$, $5^0$,
$5^0$ and there are only two unique arrangements of these two sets