Season 6 Episode 1
Keep rolling a die and keep track of the running total. Are some numbers more likely to appear than others? We investigate in this episode of the maths club!
Further Reading
Recurrence relations
Sometimes you have a sequence that you’d like to work out, and you know the first few values, and you know each value in terms of the previous values. That’s OK because then you can work out the terms one-by-one. The equation relating together different values in your sequence is called a recurrence relation. You can learn about them on Bitesize.
We once tackled some more complicated recursion problems on the MAT Livestream.
Proof by Induction
There's a good introduction to proof by induction on NRICH by Vicky Neale. Once you've done one proof by induction, you'll find that you can do the next one in the same manner. That article includes some exercises, and then here are some more, also on NRICH;
See also; the questions 94-S1-Q7 and 17-S1-Q8 on the STEP database.
Evil numbers
For more discussion of the sums of decimal digits of a number, see Wolfram Mathworld.
There’s another definition of evil numbers that has nothing to do with 666 which I should mention here. It comes from combinatorial game theory. The book Winning Ways for Your Mathematical Plays Volume 3 (Berlekamp, Conway, Guy) defines the following;
- Evil numbers are nonnegative integers with an even number of 1s in their binary expansion.
- Odious numbers are nonnegative integers with an odd number of 1s in their binary expansion.
These are relevant for some Nim-like games, and this is the definition of "evil number" that Wikipedia has an article for
Numberphile has an episode titled The Most Evil Number which they released on Halloween a few years ago which is entirely unrelated but still fun.
Large dice
Towards the end of the livestream I got distracted by experimenting with a very large number of rolls and a very large number of sides of the die. In this case, the shape of the graph takes on a particular form. I reckon that there are some equations involving integrals that describe this curve.
We can turn this into a sort of puzzle.
"A dragon has trapped you in a room with an $N$-sided die. The integer $N$ is very large. The dragon is also very large. The dragon tells you that must choose a target number, and then each day you will roll the die until either the total of your rolls is equal to that target, or until the total exceeds the target number. If you reach precisely your target number then you can leave, but if you go over the target you must try again the next day. To stop you from getting lucky on your first roll, the dragon insists that your target number must be larger than $N$. What target number (roughly) should you choose?"
Warning: I think that this puzzle is very difficult!
The word "roughly" is there because I think determining the exact best number in terms of $N$ is extraordinarily difficult.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.