I have another blog that doesn't suck. Archive:
Comments disabled 
An earlier shitty post on this shitty blog offered a simple method for proving that !!2^{50} < 3^{33}!! without calculating the values:
Katara very correctly observed that this is incomplete, because it may not be clear how to carry out the instruction “take a heap of red beans of size…”. But there is a solution! Start with a heap of 32 red beans. Now double the size of the heap by adding one bean for each bean already in the heap; this can be done without any calculation or even counting:
You now have a heap of !!2^6!! beans. Repeat this doubling process 44 more times. One can similarly construct a heap of !!3^{33}!! navy beans without calculation. This assumes that we can count as far as 44, which I think is not too unreasonable; I myself can usually get as far as 50 or even 65 on a good day. But if not, workarounds are available. I will continue to resist the temptation to suggest that there is a deep and subtle point lurking here.
