Easily formulated, and superficially similar questions 
on prime numbers, can in fact range from the very easy 
to the extraordinarily difficult, quite unexpectedly.

As an example, arrange the positive integers into four 
columns, as follows:

       1     2*    3*    4
       5*    6     7*    8
       9    10    11*   12
      13*   14    15    16
      17*   18    19*   20
      21    22    23*   24
      25    26    27    28
      29*   39    31*   32
      33    34    35    36
      37*   38    39    40
      41*   42    43*   44
      45    46    47*   48
      49    50    51    52
      53*   54    55    56
      57    58    59*   60
      61*   62    63    64
      65    66    67*   68
      69    70    71*   72
      73*   74    75    76
      77    78    79*   80
      81    82    83*   84
      85    86    87    88
      89*   90    91    92
      93    94    95    96
      97*   98    99   100
       :     :     :     :
       :     :     :     :

A primary school child can undestand why there is just 
one prime in column 2, and no primes in column 4.

A secondary school student can understand Euclid's proof 
that columns 1 and 3, taken together, contain infinitely 
many primes.

An undergraduate student can understand why column 1 
contains infinitely many primes (a theorem proved 
in this course), or why there must be infinitely 
many ROWS with NO primes.

A postgraduate student can understand why `half' of the 
primes are in column 1 and `half' in column 3 (in a 
the sense that can be made very precise).

Many mathematicians believe that there are infinitely 
many ROWS containing TWO primes, but nobody has been 
able to prove it.