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.