[Guest]

We have a number. All the digits of this number are different from one another and all the digits except the first and the last have a greater value than the mean of its neighbors (i.e. the digits immediately to the left and to the right).

What can this number be at maximum?

[Guest]

9865320

[Guest]

9865320 is unfortunately wrong

the mean of 8,5 is 6.5>6

and the mean of 5,2 is 3.5>3

I think the greatest number won't exceed 6 digits

the maximum that I got is

689740

[Guest]

I found a 8 digit number:

36899863

Edited August 26, 2010 by Jobe17

[Guest]

9863 is the greatest valid number.

[Guest]

Sorry ifound a greater number:

3689740

[Guest]

I came up with 89740, but I see now I didn't think about adding some of the numbers I had left to the front end.

[Guest]

3689740

Edited August 26, 2010 by DaTree

[Guest]

I too got 3,689,740. But thinking outside the box, if you are looking for the greatest magnitude and you let the - sign fall on the 1 when comparing neighbor averages you get -13,689,740.

[Guest]

I found a 8 digit number:

36899863

So I completely missed the part about all the digits needing to be different. With that in mind, I now get the following number:

3689740

Just out of curiosity, what would be the largest number be if duplicate digits were allowed? See the first spoiler in this post for my answer.

[Guest]

All but one person have forgotten the main caveat of the puzzle: "the digits immediately to the left and to the right"

I take this to mean one digit from the left side and one digit from the right side. That being said, a 9 can be nowhere in the number except at the beginning or end.

9863

[Guest]

The largest base number in the decimal (base-10) system is 9863.

Even larger numbers can be expressed if the size of the base is increased, such as base-16, or if the digit(s) at the end of the number could be raised as an exponent (for example 986³).

Edited August 27, 2010 by Dej Mar

[Guest]

Ignore my last post. Was in the middle of correcting it when accidently pasted the incorrect text...then unfortunately time ran out when I was editting it.

The largest base number in the decimal (base-10) system is 3689740.

Even larger numbers can be expressed if the size of the base is increased, such as base-16, or if the digit(s) at the end of the number could be lowered as a base representation of the number (for example 36897₄₀) or raised as an exponent (for example 36897⁴⁰).

Edited August 27, 2010 by Dej Mar

[Guest]

I got the same as the others.

Here's my logic.

For any digit, x, let's call it's neighbors

x+y and x+z

the mean of these is x + (y+z)/2

this is less than x when

y+z < 0

That is, the sum of differences of going from the current digit to either of it's neighbors must be negative overall.

The highest digit is thus kind of like the top of a mountain, with all other digits decreasing as you head in either direction from that position.

So, 9 is the top of this mountain.

It's neighbors must be lower than it, but we don't want them to be that much lower than it because slower decrease means more digits means larger overall number.

So it is logical to say that the neighbors are 8 and 7.

2 choices then, 897 and 798.

Choose the bigger one.

897

Now, 8 needs another neighbor such that the sum of differences between the neighbors and 8 is negative.

on the right side of 8 is a +1 to get to 9.

This means that the left side of 8 must at least be a -2 or lower.

A -2 gives 6

6897

On the right side for the 7, we already have a +2 to go from 7 to 9, so we need at least a -3

68974

On the left side we have a +2, so we need a -3

368974

On the right side we have a +3, so we need a -4

3689740

voila

Edited August 27, 2010 by mmiguel1

[Guest]

We have a number. All the digits of this number are different from one another and all the digits except the first and the last have a greater value than the mean of its neighbors (i.e. the digits immediately to the left and to the right).

What can this number be at maximum?

3689863

[Guest]

3689863

You used 8 twice.

[Guest]

368752