Eric Angelini wrote:

Show once and only once every couple of digits 00, 01, 02, 03, ..., 97, 98, 99 in the lightest finite strictly increasing sequence of integers. ["Weight" of a sequence is given by the sum of all a(n)]

Nice problem! Assuming you are not allowed to write numbers with leading zeros (like "05"), here's my best attempt: 20 23 24 25 26 27 28 29 30 34 35 36 37 38 39 40 45 46 47 48 49 50 56 57 58 59 60 67 68 69 70 78 79 80 89 90 91 100 122 133 144 155 166 177 188 192 It has weight 3224. The pairs "11", "22", "33"... are troublesome because if you try to cover them using numbers less than 100 you find yourself with very little room to manouevre. I'd love to be able to say something clever about a lower bound for the weight of such sequences... but I haven't been able to prove anything. r/