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Problem 220

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Heighway Dragon

Let D₀ be the two-letter string “Fa”. For n≥1, derive D_n from D_n-1 by the string-rewriting rules:

“a” → “aRbFR”
“b” → “LFaLb”

Thus, D₀ = “Fa”, D₁ = “FaRbFR”, D₂ = “FaRbFRRLFaLbFR”, and so on.

These strings can be interpreted as instructions to a computer graphics program, with “F” meaning “draw forward one unit”, “L” meaning “turn left 90 degrees”, “R” meaning “turn right 90 degrees”, and “a” and “b” being ignored. The initial position of the computer cursor is (0,0), pointing up towards (0,1).

Then D_n is an exotic drawing known as the Heighway Dragon of order n. For example, D₁₀ is shown below; counting each “F” as one step, the highlighted spot at (18,16) is the position reached after 500 steps.

What is the position of the cursor after 10¹² steps in D₅₀ ?
Give your answer in the form x,y with no spaces.

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海韦龙形曲线

记D₀为包含两个字符的字符串“Fa”。对于n≥1，我们通过如下的字符串重写规则，由D_n-1推出D_n：

“a” → “aRbFR”
“b” → “LFaLb”

因此，D₀ = “Fa”，D₁ = “FaRbFR”，D₂ = “FaRbFRRLFaLbFR”，依此类推。

这些字符串可以作为一个计算机绘图程序的指令加以解读，其中“F”表示“向前画一单位”，“L”表示“左转90度”，“R”表示“右转90度”，而“a”和“b”可以忽略。初始状态下，计算机光标的位置是(0,0)，方向指向(0,1)。

根据D_n绘制的奇异图象被称为n阶海韦龙形曲线。例如，如下所示为D₁₀；如果把每个“F”算作一步，高亮的点(18,16)是经过500步时所到达的位置。

根据D₅₀绘制图象，经过10¹²步之后，光标的位置在哪里？
将你的答案写作“x,y”的形式，中间没有空格。

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