Continuous Everywhere but Differentiable Nowhere

The 6th Monday Math Madness is online now. This week is actually not so hard, even though there are two different problems… I was able to get both of the answers in about 30 minutes (assuming I didn’t make any huge errors). I especially like the first question, because it can be easily transposed into a slightly more difficult and fun problem…

The first question reads:

• Start with 500 gallons of mayonaise.
  1) Mix in 10 gallons of ketchup. Stir until completely mixed.
  2) Remove 10 gallons of the mixture.
  3) Repeat steps 1 and 2 until the mixture is approximately 50% mayonaise and 50% ketchup.  

  How many iterations will it take to do this?

Here’s the slightly modified problem:

• Start with 500 gallons of mayonaise.
  1) Mix in 3 gallons of mayonaise and 7 gallons of ketchup. Stir until completely mixed.
  2) Remove 10 gallons of the mixture.
  3) Repeat steps 1 and 2 until the mixture is approximately 40% mayonaise and 60% ketchup.  

  How many iterations will it take to do this? 

That slight change makes it more difficult! But fun fun fun!