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 2019-12-27 
In tonight's Royal Institution Christmas lecture, Hannah Fry and Matt Parker demonstrated how machine learning works using MENACE.
The copy of MENACE that appeared in the lecture was build and trained by me. During the training, I logged all the moved made by MENACE and the humans playing against them, and using this data I have created some visualisations of the machine's learning.
First up, here's a visualisation of the likelihood of MENACE choosing different moves as they play games. The thickness of each arrow represented the number of beads in the box corresponding to that move, so thicker arrows represent more likely moves.
The likelihood that MENACE will play each move.
There's an awful lot of arrows in this diagram, so it's clearer if we just visualise a few boxes. This animation shows how the number of beads in the first box changes over time.
The beads in the first box.
You can see that MENACE learnt that they should always play in the centre first, an ends up with a large number of green beads and almost none of the other colours. The following animations show the number of beads changing in some other boxes.
MENACE learns that the top left is a good move.
MENACE learns that the middle right is a good move.
MENACE is very likely to draw from this position so learns that almost all the possible moves are good moves.
The numbers in these change less often, as they are not used in every game: they are only used when the game reached the positions shown on the boxes.
We can visualise MENACE's learning progress by plotting how the number of beads in the first box changes over time.
The number of beads in MENACE's first box.
Alternatively, we could plot how the number of wins, loses and draws changes over time or view this as an animated bar chart.
The number of games MENACE wins, loses and draws.
The number of games MENACE has won, lost and drawn.
If you have any ideas for other interesting ways to present this data, let me know in the comments below.

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Comments

Comments in green were written by me. Comments in blue were not written by me.
@(anonymous): Have you been refreshing the page? Every time you refresh it resets MENACE to before it has learnt anything.

It takes around 80 games for MENACE to learn against the perfect AI. So it could be you've not left it playing for long enough? (Try turning the speed up to watch MENACE get better.)
Matthew
                 Reply
I have played around menace a bit and frankly it doesnt seem to be learning i occasionally play with it and it draws but againt the perfect ai you dont see as many draws, the perfect ai wins alot more
(anonymous)
                 Reply
@Colin: You can set MENACE playing against MENACE2 (MENACE that plays second) on the interactive MENACE. MENACE2's starting numbers of beads and incentives may need some tweaking to give it a chance though; I've been meaning to look into this in more detail at some point...
Matthew
                 Reply
Idle pondering (and something you may have covered elsewhere): what's the evolution as MENACE plays against itself? (Assuming MENACE can play both sides.)
Colin
                 Reply
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 2018-06-16 
This year, like every World Cup year, I've been collecting stickers to fill the official Panini World Cup sticker album. Back in March, I calculated that I should expect it to cost £268.99 to fill this year's album (if I order the last 50 stickers). As of 6pm yesterday, I need 47 stickers to complete the album (and have placed an order on the Panini website for these).

So... How much did it cost?

In total, I have bought 1781 stickers (including the 47 I ordered) at a cost of £275.93. The plot below shows the money spent against the number of stickers stuck in, compared with the what I predicted in March.
To create this plot, I've been keeping track of exactly which stickers were in each pack I bought. Using this data, we can look for a few more things. If you want to play with the data yourself, there's a link at the bottom to download it.

Swaps

The bar chart below shows the number of copies of each sticker I got (excluding the 47 that I ordered). Unsurprisingly, it looks a lot like random noise.
The sticker I got most copies of was sticker 545, showing Panana player Armando Cooper.
Armando Cooper on sticker 545
I got swaps of 513 different stickers, meaning I'm only 169 stickers short of filling a second album.

First pack of all swaps

Everyone who has every done a sticker book will remember the awful feeling you get when you first get a pack of all swaps. For me, the first time this happened was the 50th pack. The plot below shows when the first pack of all swaps occurred in 500,000 simulations.
Looks like I was really quite unlucky to get a pack of all swaps so soon.

Duplicates in a pack

In all the 345 packs that I bought, there wasn't a single pack that contained two copies of the same sticker. In fact, I don't remember ever getting two of the same sticker in a pack. For a while I've been wondering if this is because Panini ensure that packs don't contain duplicates, or if it's simply very unlikely that they do.
If it was down to unlikeliness, the probability of having no duplicates in one pack would be:
\begin{align} \mathbb{P}(\text{no duplicates in a pack}) &= 1 \times\frac{681}{682}\times\frac{680}{682}\times\frac{679}{682}\times\frac{678}{682}\\ &= 0.985 \end{align}
and the probability of none of my 345 containing a duplicate would be:
\begin{align} \mathbb{P}(\text{no duplicates in 345 packs}) &= 0.985^{345}\\ &= 0.00628 \end{align}
This is very very small, so it's safe to conclude that Panini do indeed ensure that packs do not contain duplicates.

The data

If you'd like to have a play with the data yourself, you can download it here. Let me know if you do anything with it...

Similar posts

World Cup stickers 2018, pt. 3
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