# Satanic numbers

1. Apr 6, 2004

### EL

Could someone please help me with this problem which I have suffered many hours in front of? I never came to any solution which totally satisfied me, and I'm guessing I'm on the wrong track...

A number x in the interval [0,1] is called "satanic" if the decimal expansion of x contains somewhere the sequence 666.
Show that "almost all" numbers in [0,1] are satanic, i.e., that m([0,1]\S)=0 where S is the set of satanic numbers, and m is the Lebesgue measure.

2. Apr 6, 2004

### matt grime

Let us assume [0,1]\S is measurable (I can't see why it shouldn't be), its measure is then the probability that a number I pick at random from [0,1] doesn't contain 666 at any point in its expansion. But that probability is zero, as the 666 occurs roughly once in every 1000 blocks of three digits, so the probablity it doesn't occur in the first n*1000 places is 999/1000)^n which tends to zero as n tends to infinity.