Circle of Lamps

In a circle of lamps, initially all the lamps are on. Go clockwise around the circle, examine each lamp, and change the state of the next lamp according to the following rule. If the lamp being examined is on, then the next lamp's state is flipped (switch the lamp from on to off, or vice versa). If the lamp being examined is off, then the next lamp remains as it was. The number in the middle is the step number.
The first question is: for which values of will it ever again happen that all lamps are (simultaneously) on?
The second question is: if such a value of is found, can you derive a general formula (a function of ) for the smallest (positive) number of steps to get back to the initial state?


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This Demonstration was inspired by Problem 6 from [1].
The arrow points to the currently examined lamp. The "next" button steps to the next state. The "automatic" button steps automatically until returning to the current state or the all-on state (whichever comes first). The "reset" button switches all lamps on and examines the lamp at the top. Click the "reveal" button to change it into a button that serves as a hint for answering the first question. You can also change the state of each lamp by clicking it. The step counter can be reset to zero by clicking it.
To answer the second question, the problem can be modeled as a linear feedback shift register or a cellular automaton, with binary flip-flops/cells; also as a polynomial over GF(2) (the binary field), that is, a polynomial with binary coefficients of degree where the next state is obtained by multiplying the polynomial by and taking the result modulo (the so-called characteristic polynomial of this update process; it has degree ).
For more information on polynomials over finite fields and linear feedback shift registers, see [2].
The theory tells us that because the characteristic polynomial does not vanish for , each state belongs to a unique cycle, because each state has a unique preceding state. The cycle with the longest period is the one with the state 1, and this state evolves into the all-on state. That longest period is equal to the so-called polynomial order of the characteristic polynomial, which is the smallest positive exponent such that equals 1 modulo the characteristic polynomial. This number can be calculated (see the "information" button). Other cycles have a period that is a divisor of this longest period. In case the characteristic polynomial is primitive, the longest period equals . Note that 0 (all off) always forms a cycle of length 1 on its own.
For example, in the case of , we have that the characteristic polynomial factors as , where both factors are primitive, having longest periods 3 and 7, respectively. Therefore, the period for equals . There are also three other cycles: containing (period 7), (period 3), and 0 (period 1).
Warning: the computation of the polynomial order takes a while for and .
[1] International Mathematical Olympiad. "34th IMO 1993."
[2] R. Lidl and H. Niederreiter, Chapters 3 and 6, Introduction to Finite Fields and Their Applications, New York: Cambridge University Press, 1994.
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