BMAD

I think you are on the right track rocdocmac

you have not proven nor disproven this

I have in mind a number which, when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 2. Am I telling the truth?

Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?

You have the right idea with your comment. I an designing a cut as a swipe of the knife, so lining them up and then cutting with one swipe would count as one cut.

I wish to share 30 identical individual sausages equally amongst 18 people. What is the minimum number of cuts I need to make? What is the minimum number of pieces I need to create?

Assume that only four buttons work on your calculator: 5, 7, enter, and plus. What whole numbers can you not use your calculator to make?

yes, this is what I mean.

James and Mike are running in a race. They both walked and ran for part of the rate. They each walked and ran at the same speed. James ran for half the distance and walked for half the distance. Mike ran for half of his time and walked for half of his time. Who finished first?

All you showed is the bound is wider than one thought.

So then if we know N and P we should be able to bound squares x by a two values. Are those values always consecutive?

I agree

Find a function where the arc lenth and area between any two randomly defined points is the same. There are two.

Say we have the function: y=x^x^x^x^x..... Find an x value for which the derivative of this function converges. If you are really clever you'll find the interval that converges.