The answer is no. In fact, if, at any stage,
all the coins in at least one circle are heads, it's impossible to proceed to the desired final configuration by means of FLIPS and CLEARS. This not only covers a wide range of initial conditions, it also implies that for arbitrary initial configurations no sequence of operations containing a CLEAR can yield the desired final configuration. PROOF: Assume that at some stage (possibly the initial config- uration) all the coins in at least one circle are heads. Therefore, any sequence of FLIPS and CLEARS can be assumed to contain at least one CLEAR. Let A denote the circle that is the subject of the last CLEAR in the sequence, and let B and C denote the other two circles. Every operation following the last CLEAR is a FLIP of either A, B, or C. Let nA, nB, and nC denote the number of these FLIPS respect- ively. Since the coin in the triple-region was turned heads up by the last CLEAR, the sum (nA+nB+nC) must be odd for the triple-region to end up tails. The double-regions of AB and AC were also turned heads up by the last CLEAR, so the sums (nA+nB) and (nA+nC) must both be even to leave these regions heads up. It follows that nA, nB, and nC must each be odd. However, the oddness of nA implies that the single-region of A, which was turned heads up by the last CLEAR, ends up tails. Thus, the desired final configuration cannot be reached. |
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