1. ## set problem

Let Si be the set of all intergers n such that 100i < or = n < 100 (i + 1).
For example, S4 is the set {400, 401,402,........499}. How many of the sets S0, S1, S2, S3,................ S999 do not contain a perfect square?

I tried to make a pattern
1 2 3 4 5 6 7 8 ............ 50
1 4 9 16 25 36 49 64 ............ 2500
3 5 7 9 11 13 15
and I see the difference between two squares have a pattern of 2n-1

therefore, when 2n-1 >100 there will be some sets that don't have a square. n <50 there will be squares in each set.
Up to S25 there should be at least one square in each set.

I am stuck from here because it will be impossible for me to try out all the sets...

Thanks.

Vicky.

2. Originally Posted by Vicky1997
Let Si be the set of all intergers n such that 100i < or = n < 100 (i + 1).
For example, S4 is the set {400, 401,402,........499}. How many of the sets S0 S1 S2 S3,................ S999 do not contain a perfect square?
I do not understand this question as posted.
The fact is that each set $S_i$ as defined contains a square.
Are you asking "how many subsets of $S_n$ do not contain a square?"

3. Originally Posted by Plato
I do not understand this question as posted.
The fact is that each set $S_i$ as defined contains a square.
Are you asking "how many subsets of $S_n$ do not contain a square?"
I was trying to find the sets that do not contain a square.
For example, I think S91 does not contain a square because the square of 95 is 9025 and the square of 96 is 9216.

Vicky.

4. Originally Posted by Vicky1997
Let Si be the set of all intergers n such that 100i < or = n < 100 (i + 1).
For example, S4 is the set {400, 401,402,........499}. How many of the sets S0, S1, S2, S3,................ S999 do not contain a perfect square?

I tried to make a pattern
1 2 3 4 5 6 7 8 ............ 50
1 4 9 16 25 36 49 64 ............ 2500
3 5 7 9 11 13 15
and I see the difference between two squares have a pattern of 2n-1

therefore, when 2n-1 >100 there will be some sets that don't have a square. n <50 there will be squares in each set.
Up to S25 there should be at least one square in each set.

I am stuck from here because it will be impossible for me to try out all the sets...

Thanks.

Vicky.

Of course, it's impossible to try out all the numbers. You did all the work and stopped right before you got the answer.
As you said (n+1)squared - n sqaured >100
so n>=50
50 squared is 2500 and this number is in the set S25.
S0 to S25 all have at least one square.

Now let's look at the sets S26 to S999.
Most of the sets will not have a square and the ones that do can't have more than one. This fact makes the answer obvious.

The biggest number is the set is 99999 and this is in between 316 sqaured and 317 squared. This means a total of 316 squares and we already know that 50 of them are in the sets from S0 to S25.

316 - 50 = 266
This shows that there are 266 squares in the 974 sets from S26 to S999.
974-266 = 708

The answer should be 708. I went over my work 5 times to make sure there are no errors in my calculation.

If you still haven't solved this problem,