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" Several other states are exploring the sale or long-term lease of their lotteries . By adjusting coefficients in the Turing model 's equations , it 's possible to vary the initial distribution of the hypothetical morphogens that mix together to create the colors and produce swirls , spots , or stripes . In 1927 , van der Waerden proved the following generalization : If n is a sufficiently large integer and if each integer from 1 through n is printed arbitrarily in one of two colors , there is always a monochromatic arithmetic progression with a certain number of terms . Turing machines are restricted to a finite number of states . It manifests itself in the form of clusters of panels hanging on the walls of recording studios , concert halls , nightclubs , and other venues . Fighting back lately , Giuliani has started to sound and act like a strong contender , traveling to the early primary states of New Hampshire and South Carolina , and arguing that his vision for the future and performance in the past would make him a formidable GOP nominee . Suppose that each integer from 1 through 9 is printed on a page in one of two colors , either red or blue ( shown below in bold ) . She also enjoyed an art course that involved going to New York City to visit galleries , museums , and artist 's studios . In general , for a given number of colors , the idea is to identify arithmetic progressions in which each integer has a different color . " To study the laws of history we must completely change the subject of our observation , must leave aside kings , ministers , and generals , and study the common , infinitesimally small elements by which the masses are moved , " Tolstoy wrote in War and Peace . At least 17 other states are debating whether to require the vaccine , including California , New Jersey and Virginia , as well as the District of Columbia .