Gravitational lensing can amplify the apparent brightness of distant sources .
Images that are highly magnified are often part of multiply-imaged systems , but we consider the possibility of having large magnifications without additional detectable images .
In rare but non-negligible situations , lensing can produce a singly highly magnified image ; this phenomenon is mainly associated with massive cluster-scale halos ( \gtrsim 10 ^ { 13.5 } M _ { \odot } ) .
Alternatively , lensing can produce multiply-imaged systems in which the extra images are either unresolved or too faint to be detectable .
This phenomenon is dominated by galaxies and lower-mass halos ( \lesssim 10 ^ { 12 } M _ { \odot } ) , and is very sensitive to the inner density profile of the halos .
Although we study the general problem , we customize our calculations to four quasars at redshift z \approx 6 in the Sloan Digital Sky Survey ( SDSS ) , for which ( ) 
have ruled out the presence of extra images down to an image splitting of \Delta \theta = 0 \farcs 3 and a flux ratio of f = 0.01 .
We predict that 9–29 % of all z \approx 6 quasars that are magnified by a factor of \mu > 10 would lack detectable extra images , with 5–10 % being true singly-imaged systems .
The maximum of 29 % is reached only in the unlikely event that all low-mass ( \lesssim 10 ^ { 10 } M _ { \odot } ) halos have highly concentrated ( isothermal ) profiles .
In more realistic models where dwarf halos have flatter ( NFW ) inner profiles , the maximum probability is \sim 10 % .
We conclude that the probability that all four SDSS quasars are magnified by a factor of 10 is \lesssim 10 ^ { -4 } .
The only escape from this conclusion is if there are many ( > 10 ) multiply-imaged z \approx 6 quasars in the SDSS database that have not yet been identified , which seems unlikely .
In other words , lensing can not explain the brightnesses of the z \approx 6 quasars , and models that invoke lensing to avoid having billion- M _ { \odot } black holes in the young universe are not viable .