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Intensive experiments have revealed that the superconductivity of thehole-doped cuprates can be strongly suppressed at the so-called magic dopingfractions. Despite great research efforts, the origin of the `magic doping'remains mysterious. Recently, we have developed a real-space theory ofhigh-temperature superconductivity which reveals the intrinsic relationshipbetween the localized Cooper pair and the localized hole pair(arXiv:1007.3536). Here we report that the theory can naturally explain theemergence of non-superconducting checkerboard phases and the magic dopingproblem in hole-doped cuprate superconductors. It clearly shows that thereexist only seven `magic numbers' in the cuprate family at $x=$1/18, 1/16, 2/25,1/9, 1/8, 2/9 and 1/4 with 6a*6a, 4a*4a, 5a*5a, 3a*3a, 4a*4a, 3a*3a and 2a*2acheckerboard patterns, respectively. Moreover, our framework leads directly toa satisfactory explanation of the most recent discovery [M. J. Lawler, et al.Nature 466, 347 (2010)] of the symmetries broken within each copper-oxide unitin hole-doped cuprate superconductors. These findings may shed new light on themechanism of superconductivity.

Magic doping: From the localized hole-pair to the checkerboard patterns