ducci | zeracuse

It’s a bit tricky such and a beauty too:

Choose 4 real numbers $a,b,c,d$ . Write down:

$a_{1}=a, b_{1}=b, c_{1}=c, d_{1}=d$ .

In the following line write the new set: $a_{2}, b_{2}, c_{2}, d_{2}$ where:

$a_{2} = \left|a_{1}-b_{1}\right|,b_{2} = \left|b_{1}-c_{1}\right|,c_{2} = \left|c_{1}-d_{1}\right|,d_{2} = \left|d_{1}-a_{1}\right|$ Continuing this procedure prove that after some $k$ steps we will take:

$\left(a_{k},b_{k},c_{k},d_{k}\right) = \left(0,0,0,0\right)$

Note: This sequence is called ducci.

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find the invariant