Cut the Cards

[Albert Chan]

Here is simulations, using Lua
Note: math.random(n) return 1 .. n, each with same probability.

Code:

function run(tests,cards)   -- averaged picks
    local n = 0             -- bad tries
    for i=1,tests do
        for t=1,cards do
            while math.random(cards) < t do n=n+1 end
        end
    end
    return n/tests + cards
end

lua> run(1e6, 52)
235.970842