How many sequences of coin flips of length \(n\) do not contain \(HH\)? E.g. there are 5 for \(n = 3: TTT, TTH, THT, HTT, HTH\).

In the game Chomp, two players begin with a rectangular grid. Player 1 chooses any square and removes it from the grid, together with all the squares above and to the right of it. Player 2 chooses one of the remaining squares and removes that, together with all the squares above it and to its right. The two take turns in this way until one of them is forced to remove the last square. That player loses.

If the starting grid is square, can either player force a win?

Particle \(P_0\) is at \((0, 0)\) and particle \(P_1\) is at \((0, 1)\) at time \(t=0\). \(P_0\) moves right along the \(x\)-axis at speed \(1\) and \(P_1\) always moves directly towards \(P_0\), also at speed \(1\). In the long run, \(P_1\) will be moving right along the \(x\)-axis at speed \(1\), and so the distance between the two particles will no longer decrease. What is that final distance?

@mscroggs Here are all the right-angled triangles for question 12 of the advent calendar.

@mscroggs \(\sum_{4 = 1}^\infty 4P(6 = 4 \mid even) P(even)\)

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\[\sqrt{\sqrt{9} - \sqrt{8}} = \sqrt{2} - \sqrt{1}\]


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