Arithmetic mean: every figure but 1st, last, is smaller than arith. mean of previous,

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broccoli

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Hello everyone,

first of all, I'd like to apologise for this not-so-well chosen title but I couldn't come up with anything else.
Today I was given some tricky maths "problem" and I'm kind of stuck at the moment so I hope you guys can help me:

"What is the highest natural number which has the characteristic that every figure except for the first and the last one is smaller than the arithmetic mean of the previous and following number."

I already managed to put this question into some proper statements.

I. The number of figures of the searched number must be equal to or greater than the number of figures of the previous number

II. The number of figures of the following number must be equal to or greater than the number of figures of the searched number

III. and obviously the arithmetic mean of the following and previous number must be higher than every figure of the searched number expect for the first and last one.

Have you got any suggestions on how I could solve this task? I'm explicitly just asking for hints.?

Thank you very much in advance. :)
 
Hello everyone,

I'm stuck at the following question:

What is the highest natural number which has the characteristic that every figure except for the first and the last one is smaller than the arithmetic mean of the previous and following number.

All I've got so far are some short statements/assumptions that summaries this question.

I. The searched number is n-figured. The following one is n or (n+1)-figured, the previous one is n or (n-1)-figured
II. The arithmetic mean of the previous and following number is greater than every figure of the searched number except for the first and the last one

I'm explicitly just asking for a hint.?

Thank you very much in advance
 
I'm not sure the question is well-defined. Maybe it is.

Can you offer three numbers, explain the test (criterion, calculation), and tell why the number passes or fails? We can search for the greatest, later. Let's make sure we understand the mechanics.
 
broccoli said:
What is the highest natural number which has the characteristic that every figure except for the first and the last one is smaller than the arithmetic mean of the previous and following number.

All I've got so far are some short statements/assumptions that summaries this question.

I. The searched number is n-figured. The following one is n or (n+1)-figured, the previous one is n or (n-1)-figured
II. The arithmetic mean of the previous and following number is greater than every figure of the searched number except for the first and the last one
Click to expand...

I'm trying to interpret the problem, and the best I can do seems to conflict with your thoughts. The issue is the meaning of "figure" and "number".

I take "figure" to mean one of the digits making up the number (in base 10?). But that would imply that "number" means something else - either the entire number, or perhaps the numbers made up of the digits before and after a given digit.

But the problem only makes sense to me if I suppose that was a mistake, and "figure" was meant:

What is the highest natural number which has the characteristic that every figure except for the first and the last one is smaller than the arithmetic mean of the previous and following figures.
​

Then, for example, the number 124 would work, because 2 is less than the average of 1 and 4 (namely 3).

This way, we aren't comparing a single digit with a multi-digit number; the former would always be smaller, so that condition is trivial.

Is that a reasonable interpretation of the problem?
 
Please don't post twice. Be patient while your post is in moderation, if that's what happened.

In the other thread, we are asking for clarification. Please do so there.

As I suggested there, I think you are misunderstanding the problem. Did you quote it exactly as given to you?
 
Thank you for the quick replies! Unfortunately, the question is meant the way I put it.

I'll try to give an example:

Let's say the searched number was 1000. The previous number is 999, the following is 1001.
The arithmetic mean of 999 is (9+9+9)/3= 9. The arithmetic mean of 1001 is (1+0+0+1)/4=0.5
Therefore the arithmetic mean of the previous number and the following number is greater than the figures of the searched number except for the first and the last one -> (1)00(0) -> 9 and 3 is greater than 0 and 0.

The thing is though that I've got to find the greatest natural number which has got this "characteristic".
 
I think I followed you until you said 3 > 0. Did you mean [9 > 0 and 0] and [½ > 0 and 0]?

New Sample

Search: 12345
Before: 12344
After: 12346

A.Mean.Before (1+2+3+4+4)/5 = 2.8
A.Mean.After (1+2+3+4+6)/4 = 3.2

(1)234(5) --> Consider 2, 3, and 4

2.8 > 2 - Good
2.8 < 3 - Fail
2.8 < 4 - Fail

3.2 > 2 - Good
3.2 > 3 - Good
3.2 < 4 - Fail

Are we on the same page?
 
Last edited:
broccoli said:
Thank you for the quick replies! Unfortunately, the question is meant the way I put it.

I'll try to give an example:

Let's say the searched number was 1000. The previous number is 999, the following is 1001.
The arithmetic mean of 999 is (9+9+9)/3= 9. The arithmetic mean of 1001 is (1+0+0+1)/4=0.5
Therefore the arithmetic mean of the previous number and the following number is greater than the figures of the searched number except for the first and the last one -> (1)00(0) -> 9 and 3 is greater than 0 and 0.

The thing is though that I've got to find the greatest natural number which has got this "characteristic".
Click to expand...

Well, the way you put it was, "every figure except for the first and the last one is smaller than the arithmetic mean of the previous and following number". The arithmetic mean of 999 and 1001 is 1000! Apparently you meant "the arithmetic mean of the digits of either the previous or the following number". That's very different.

Is what you stated the exact wording of the problem as given to you? And where did it come from?
 
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