What to do (3)

Following on from last Tuesday's post - if you're going to use X (or any other letter or symbol) to represent an unknown, then you have to be consistent - it must represent the same unknown throughout - and it is here that our reasoning falls apart.  In our initial statement - that if our envelope contains X, the other envelope must contain either 2X or X/2 - we are using X to represent two different values - firstly (the 2X bit) the lesser of the two amounts on offer and then (X/2) the greater of the two.  It becomes more obvious if we use real money - let's suppose that we know that one envelope contains £10 and the other one £20.  If we have chosen the envelope with £10 in it (i.e. X = 10) then the other envelope must contain £20 (2X). It cannot contain X/2 (£5).  Equally if our envelope contains £20, the other must contain £10 (X/2).  It cannot contain 2X (£40).  So our initial statement is false, and everything which flows from it is equally false.  If X represents the smaller amount (£10 in our example) then clearly whichever envelope we choose, the other envelope must contain either 2X or X and as that goes for either envelope, there is nothing to be gained by swapping.