Don't you be no square... (A 10by10 square contains 100 1by1 squares...)

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Denis

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A 10by10 square contains 100 1by1 squares (of course!).
A circle is drawn inside above square, tangent to all 4 sides.
How many of the 1by1 squares are fully inside the circle?

Was fiddling with above. I get 60 as answer.
 
Denis said:
A 10by10 square contains 100 1by1 squares (of course!).
A circle is drawn inside above square, tangent to all 4 sides.
How many of the 1by1 squares are fully inside the circle?

Was fiddling with above. I get 60 as answer.
Click to expand...

I agree. There are 15 in each quadrant; the 3-4-5 right triangle helps in confirming that.
 
Thanks DrP.
Same problem with a 100by100 square?

I get 7644 (1911 per quadrant).
 
Denis said:
A 10by10 square contains 100 1by1 squares (of course!).
A circle is drawn inside above square, tangent to all 4 sides.
How many of the 1by1 squares are fully inside the circle?

Was fiddling with above. I get 60 as answer.
Click to expand...
I got just over 19pi as my answer.:smile:
 
Denis said:
A 10by10 square contains 100 1by1 squares (of course!).
A circle is drawn inside above square, tangent to all 4 sides.
How many of the 1by1 squares are fully inside the circle?

Was fiddling with above. I get 60 as answer.
Click to expand...

I disagree. To me, "fully inside the circle" means you can "move freely about inside the circle without being blocked by any part of a square."




One might ask instead "How many 1 by 1 squares don't have any of their areas outside of the circle?"
 
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lookagain said:
I disagree. To me, "fully inside the circle" means you can "move freely about inside the circle without being blocked by any part of a square."
Click to expand...

I don't see how you can interpret it like that; he said each square is fully inside the circle (that is, completely contained within it), and said nothing about someone moving around them, inside the circle. I also don't understand what you think is being asked for. What would your solution be?

Of course, if the wording is interpreted differently, then you are just solving a different problem; the difference is not about the math. But can you explain your interpretation of the English?
 
lookagain said:
One might ask instead "How many 1 by 1 squares don't have any of their areas outside of the circle?"
Click to expand...
That's equivalent to what I said, isn't it?

Solutions for circle's radius = 1 to 10:
1:0, 2:4, 3:16, 4:32, 5:60, 6:88, 7:120, 8:164, 9:216, 10:276

My li'l program that seems to work:
-center of circle at origin
-examine the 1/4 circle in 1st quadrant

r = radius
x,y = points on circumference

INPUT r
LOOP x FROM 1 TO r
y = SQRT(r^2 - x^2)
count = count + FLOOR(y)
ENDLOOP
PRINT count*4

If r=50, then count = 1911; 1911*4 = 7644
 
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Dr.Peterson said:
I don't see how you can interpret it like that; he said each square is fully inside the circle (that is, completely contained within it), and said nothing about someone moving around them, inside the circle. I also don't understand what you think is being asked for. What would your solution be?

Of course, if the wording is interpreted differently, then you are just solving a different problem; the difference is not about the math. But can you explain your interpretation of the English?
Click to expand...
I have been away since midnight (this location), and that is why I have not replied sooner. That's just it, if each square is "fully
inside the circle," then none of the squares that are to count may touch the circle, otherwise those squares are not fully inside the circle. Fully inside the circle squares do not intersect with the circle. If any part of a square is coincident with the circumference of that circle, then that square fails to be fully inside the circle.

The alternative stated question I gave (that Denis requoted) is the intended problem,
but I have explained the issue/difference with what was originally asked with a suggestion of a different phrase with a different meaning of what was actually intended
by the proposer.
 
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lookagain said:
That's just it, if each square is "fully inside the circle," then none of the squares that are to count may touch the circle, otherwise those squares are not fully inside the circle. Fully inside the circle squares do not intersect with the circle. If any part of a square is coincident with the circumference of that circle, then that square fails to be fully inside the circle.

The alternative stated question I gave (that Denis requoted) is the intended problem, but I have explained the issue/difference with what was originally asked with a suggestion of a different phrase with a different meaning of what was actually intended by the proposer.
Click to expand...

Okay, I get it now. You're taking "fully inside the circle" to mean "fully within the interior of the circle", rather than "fully contained by the disk consisting of the circle and its interior". Your original explanation, about a person moving around, didn't communicate that to me. And the question was not written in technical language, so a little latitude is to be expected.

On the other hand, your version, "How many 1 by 1 squares don't have any of their areas outside of the circle?", is at least as faulty, if you're looking for precise terms -- an "area" is a measurement, not a set of points!

How about, "How many 1 by 1 squares do not extend outside of the circle?"
 
lookagain said:
That's just it, if each square is "fully inside the circle," then none of the squares
that are to count may touch the circle, otherwise those squares are not
fully inside the circle.
Fully inside the circle squares do not intersect with the circle.
If any part of a square is coincident with the circumference of that circle,
then that square fails to be fully inside the circle.
Click to expand...
Thanks Lookagain.
But isn't it a fact that "lines" and "points" have no "size" in geometry?
If so, then a point on the circumference would be "inside"....I think...
If not, then I should have stated:
".....including those with a corner on the circumference" or words to that effect...
 
Denis said:
Thanks Lookagain.
But isn't it a fact that "lines" and "points" have no "size" in geometry?
If so, then a point on the circumference would be "inside"....I think...
If not, then I should have stated:
".....including those with a corner on the circumference" or words to that effect...
Click to expand...
Ah Denis forgot his real analysis somewhat. Nope, a point can lie on the boundary. If I recall correctly, a point is said to be an interior point if you find a neighborhood around that point that lies entirely within the set. A boundary point does not satisfy that condition. I finally got Denis!
 
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Dr.Peterson & my edit/emphasis said:
On the other hand, your version, "How many 1 by 1 squares don't have any of their areas outside of the circle?", is at least as faulty,
if you're looking for < < precise terms > > -- an "area" is a measurement, not a set of points!

How about, "How many 1 by 1 squares do not extend outside of the circle?" . . . . : ) : )
Click to expand...

Thank you.
 
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lookagain said:
Thank you.
Click to expand...

Actually, now that I think about it, I prefer a positive statement of the problem, something like "How many of the squares are entirely within the disk (circle together with its interior)?" The word "inside" unfortunately can be taken to mean "in the interior", as you did, so we have to avoid itm though that was not the original intent. I think "within" is okay. We could also say, "entirely contained by the disk".
 
Well, matters little to the count; only 4 outta 7644:
(x-y-r):
14-48-50
30-40-50
40-30-50
48-14-50
 
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