Originally Posted by

**enoimreh7**
Hello dear Sirs!

We try to solve this Problem, may be, you could help us to find a beginning of the solution?

Which is the biggest natural number with the quality that each of her figures is smaller except the first one and the last than the arithmetic means of her both neighbouring figures?

Remark: The correctness of the result is to be proved.

Sincerly, Enoimreh

The first step in solving a problem is to determine what it means. We recently discussed what sounds like the same problem, but stated in a way that made it impossible to be sure what it meant! This version seems much clearer, but I would clarify it slightly (as I read it) like this:

What is the greatest natural number such that each of its figures [digits], other than the first and the last, is smaller than the arithmetic

mean of its neighbouring figures?

So, first, choose a random number and show how you would *decide *whether it has the required property. Then use what you learn from doing that, to write a *small *number that has this property.

Once you've done that, we can start discussing how to find *larger *numbers that satisfy the requirement. Only then will we be able to propose and then prove the *largest*.

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