1. ## Difficult Counting Problem

What is the number of positive integers n satisfying the property that the number b of positive integers a satisfying the property that n a and
a^2 n satisfies the property that the number of possible ways we can put 3 different objects into b different boxes is at least as big as the number of
0-1 sequences of length n.

2. So I have attempted a solution... Does this make sense? The question is incredibly confusing wording but here are my thoughts to the problem...

RELEVANT EQUATIONS:
3 different objects into b different boxes = b^3
number of 0,1 sequences of length n = 2^n

SOLUTION ATTEMPT:
From the question, we know n is between a and a^2.
n is the number of positive integers
a is the positive integers before n
b is the number of a
and we know that b^3 is greater or equal to 2^n
if we try n=1, then a=1, b=1
n=2, then a=1,2, b=2
n=3, then a=1,2,3, b=3...
therefore n=b, which gives n^3 is greater than or equal to 2^n.
I don't know how to get n in other way, so I tried until n=10.
if n=10, then a=1,2,3,4,5,6,7,8,9,10, b=10
then 10^3 is not greater or equal to 2^10.
Therefore n is between 2 and 9 since if n=1, then a=1, b=1
1^3 is not greater of equal to 2^1.

3. If my understanding is correct, then n must be less than a^2. However your cases where you list the possible a's don't include this case. Therefore, although it is true for a=1, b is not equal to n.

I have done trial and error and have found the answer to be 3.

Cases:
N=1 à a <= 1 <= a^2
a = 1, b =1

N=2 à a <= 2 <= a^2
A=2, b = 1
N=3 à a <= 3 <=a^2
A = 2, 3, b=2
N=4 à a <= 4 <= a^2
A = 2,3,4 b=3
N = 5 à a <= 5 <= a^2
A= 3,4,5 b = 3
N = 6 à a <= 6 <= a^2
A = 3,4,5,6 b = 4
N = 7 à a<= 7 <= a^2
A = 3,4,5,6,7 b=5
N = 8 à a<= 8 <= a^2
A = 3,4,5,6,7,8 b = 6
N = 9 à a<= 9 <= a^2
A = 3,4,5,6,7,8,9 b= 7
N = 10 à a<=10 <= a^2
A = 4,5,6,7,8,9,10 b=7
etc...

I think this is the solution if n is between a and a^2. Does this make sense to everyone? It's late here and I may have no idea what I'm doing. What does everyone else think?