# Combinatorial circle placement

Suppose we have area bound by $0\leq x\leq 2$,$0\leq y\leq 2$

We choose 17 points, there should be at least two with distance max $d\leq\frac{1}{\sqrt{2}}$

Tried to count area of $A_{17circles}=(\frac{d}{2})^2\pi*17=6.67588438887831$

Then to take square $A_{square}=(\frac{d}{2}+2+\frac{d}{2})^2=7.32842712474619$

There is also each corner to be rounded $A_{removecorners}=d^2-(\frac{d}{2})^2\pi=0.107300918301276$

Which results in $A_{square}-A_{removecorners}=7.22112620644491$

Still $7.22112620644491>6.67588438887831$ so I cant dismiss argument that one is able to fit 17 points so, that $max_{distance}\geq d$

How to dismiss this argument? How to prove that if we put 17 points into 2 by 2 box, some of them bound to be closer than $\frac{1}{\sqrt{2}}$

Hint: Divide the $2\times 2$ square into $16$ equal little squares, by drawing three equally spaced horizontal lines, and three vertical lines. Then use the Pigeonhole Principle.
The text passage describes a combinatorial problem involving the placement of 17 points within a 2x2 square and attempts to prove that at least two points must be within a certain distance of each other. The passage includes a series of area calculations and a hint to use the Pigeonhole Principle to solve the problem.
single_question_with_answer

the passage contains a mathematical question (how to prove that if we put 17 points into a 2x2 box, some of them are bound to be closer than \( \frac{1}{\sqrt{2}} \)) and provides a hint as an answer (divide the 2x2 square into 16 equal little squares and use the pigeonhole principle).
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Combinatorial circle placement

Suppose we have area bound by $0\leq x\leq 2$,$0\leq y\leq 2$

We choose 17 points, there should be at least two with distance max $d\leq\frac{1}{\sqrt{2}}$

Tried to count area of $A_{17circles}=(\frac{d}{2})^2\pi*17=6.67588438887831$

Then to take square $A_{square}=(\frac{d}{2}+2+\frac{d}{2})^2=7.32842712474619$

There is also each corner to be rounded $A_{removecorners}=d^2-(\frac{d}{2})^2\pi=0.107300918301276$

Which results in $A_{square}-A_{removecorners}=7.22112620644491$

Still $7.22112620644491>6.67588438887831$ so I cant dismiss argument that one is able to fit 17 points so, that $max_{distance}\geq d$

How to dismiss this argument? How to prove that if we put 17 points into 2 by 2 box, some of them bound to be closer than $\frac{1}{\sqrt{2}}$

Hint: Divide the $2\times 2$ square into $16$ equal little squares, by drawing three equally spaced horizontal lines, and three vertical lines. Then use the Pigeonhole Principle.
user
The text passage describes a combinatorial problem involving the placement of 17 points within a 2x2 square and attempts to prove that at least two points must be within a certain distance of each other. The passage includes a series of area calculations and a hint to use the Pigeonhole Principle to solve the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a mathematical question (how to prove that if we put 17 points into a 2x2 box, some of them are bound to be closer than \( \frac{1}{\sqrt{2}} \)) and provides a hint as an answer (divide the 2x2 square into 16 equal little squares and use the Pigeonhole Principle).