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Colin Beveridge
A mathematician with nothing to prove. Author of and others. Blog at . Problems welcome. He/him.
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Colin Beveridge retweeted
Kevin Houston 29m
I think the main tasks now are 1. Get the appeals system in place. 2. Compensate those who have lost out. (Since the appeal system will be too late for some of them.)
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Colin Beveridge 2h
Replying to @MathsJam
The -mer in me then asks: what about Rock, Paper, Scissors, Lizard, Spock? I believe it works, but I'm going to leave that as an exercise for you to enjoy. (16/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
A reassuring consequence of this approach: since (1/3 + w/3 + w^2/3) = 0, a mixed strategy of picking each option with probability of 1/3 is unbeatable (over the long run). (15/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
Since 1 corresponds to a draw, we get a draw a quarter of the time. w represents a Charlie win, which is also a 1-in-4 shot, and Drew wins the remaining half of the games. (14/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
However, I want to have a probability distribution for the results, to I need to have (a + bw + cw^2) with a + b + c = 1. Fortunately, (1 + w + w^2) = 0, so I can add a multiple of that on to end up with (1/4 + w/4 + w^2/2). (13/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
And what's the outcome? It's the product of these two complex numbers, which works out to be w^2/4. (12/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
Similarly, if Drew played scissors half the time and the other moves a quarter each, their strategy would be 1/4 + w^2/4 + w/2. (11/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
For example, suppose Charlie played paper half the time, and the others each a quarter of the time, their strategy would be 1/4 + w/2 + w^2/4. (10/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
With this machinery in place, I can do better than represent Charlie and Drew's moves: I can represent their (fixed) mixed strategies, too! (9/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
For example: suppose Charlie plays scissors (w^2) and Drew plays paper (w^2). I multiply those together to get w^4, which is the same as w, and corresponds to 120º anticlockwise and a win for Charlie. (8/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
Rather than rotations, I can identify Charlie's possible moves of rock, paper and scissors with (respectively) 1, w and w^2, where 2w = -1 + i√3. Drew's moves are 1, w^2 and w (respectively). (7/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
The Further Mathematician in me can't see "120º rotation" without thinking (1) "HEY! Use radians like a grown-up!" and (2) "cube roots of unity!" (6/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
Turns out, it's not just the *moves* that identify with the rotations, the *results* do as well. But wait, there's more. As you may have surmised by virtue of this only being tweet 5 of 16. (5/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
Less obviously, any game that Charlie wins ends up with a net rotation of 120º anticlockwise, and any game Drew wins ends up rotating 120º clockwise. (4/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
Hopefully obviously, if the players match moves, the rotations cancel out and you end up back where you started. (3/16)
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Colin Beveridge 2h
Replying to @icecolbeveridge
For example, you might identify Charlie's moves of rock, paper and scissors with rotations of 0º, 120º and 240º anticlockwise, respectively, and Drew's moves as the same but clockwise. (2/16)
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Colin Beveridge 2h
If you'll indulge me, a thread. I was playing about with rock-paper-scissors, and I noticed something: you can model the moves as rotations. (1/16)
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Colin Beveridge retweeted
david allen green 3h
Data, datum, anecdotum, blah But am livid at concrete actual examples of state school A-Level result downgrading
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Colin Beveridge retweeted
Sam Freedman 3h
1) I'm going to try and do a fair and balanced summary thread on what's gone on today with A-levels. It is more complicated than the headlines (as ever) so please bear with me.
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Colin Beveridge retweeted
Maz Hamilton 9h
Replying to @newsmary
Imagine what would happen if we erred on the side of accidentally helping too many people, rather than not enough.
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