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Let’s name a binary cycle ${bf x}:x_1,x_2,ldots, x_{12}$ *balanced* if precisely 6 of the $x_i$‘s are 0 and 6 of them are 1. We can place the sequence round a circle and add every two adjoining digits modulo 2 to acquire a brand new binary cycle, so the brand new cycle has phrases $f({bf x}):x_1+x_2, x_2+x_3, ldots, x_{11}+x_{12}, x_{12}+x_1$ modulo 2. Now, think about the next binary cycle:

$${bf x}: 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0.$$

It occurs that $f({bf x})$ can be balanced:

$$f({bf x}):1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0,$$

and $f(f({bf x}))$ can be balanced:

$$f(f({bf x})): 0, 0, 1,0,1,0,1,1,1,0,0,1.$$

However, $f(f(f({bf x})))$ is not balanced:

$$f(f(f({bf x}))): 0,1,1,1,1,1,0,0,1,0,1,1.$$

Can you discover a binary cycle ${bf x}$ of size 12 such that ${bf x}, f({bf x}),f(f({bf x})),$ and $f(f(f({bf x})))$ are all balanced?

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