This problem has the advantage of being both easier and more interesting than the dumb combinatorics problems that one dude was posting.

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27954385)

http://asciiartist.com/wp-content/uploads/2015/01/b6973011-7dff-4e7a-a648-e42fbcfae4b3.jpg

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27954401)

*wins $1,000,000*

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27956114)

2. Assuming you mean that, then the statement is false. If you define the set of A points as {(x,y)|(x+y)mod2=0} and B points as {(x,y)|(x+y)mod2=1} where x and y are integers, then there is no A point within 1 unit of another A, and there is no B unit within another B, and there is no C unit within another C since there are no C units.

Proof that there is no A point within 1 unit of another A:

Choose two points in A, n and m. n = (x1,y1), m=(x2,y2).

The distance between m and n = sqrt((x1-x2)^2+(y1-y2)^2)).

Assume these two points are within 1 unit of each other. Because by postulation x1,y1,x2,y2 are all integers, in order to be within 1 unit of each other, either x1=x2 and |y1-y2|=1 or y1=y2 and |x1-x2|=1. The proof of this is left as an exercise for the reader, but it's pretty fucking obvious, merely tedious.

Now we defined A such that (x+y)mod2=0, so let's look at the first point, W/OLOG:

(x1+y1)mod2=0, and now do some substitution

(x2+y1)mod2=0, and by a theorem of absolute value addition I don't wnat to prove here,

(x2+(y2+1))mod2=0

x2mod2 + y2mod2 + 1mod2 = 0

By the same definition of A, we know

(x2+y2)mod2 = 0

x2mod2 + y2mod2 = 0

So combine those,

x2mod2 + y2mod2 + 1mod2 = x2mod2 + y2mod2

simplify

1mod2 = 0

1 = 0

This is a contradiction. The proof regarding two points in B is almost exactly the same and is left as an exercise for the reader.

Accordingly, the theorem has been disproved.

In fact, you can generalize the opposite statement as a theorem in N dimensions:

For any set of N-dimensional points with integral components, you can assign these points with N different labels such that no two points with the same label have a distance <= 1

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27956314)

make a circle around "A". all points in this circle must be either B or C points. wlog, choose a "B" point on the boundary of this circle. draw a circle around it. all points within this circle must be A or C points. but there is a region where the A circle and the B circle intersect. all points inside this region must be C points. but since they are all C points and they are in a continuous region where the max distance is less than one, there are two C points within distance 1.

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27956948)

But most of the points on the circle aren't 1 away from each other

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27956384)

Or did u just google the problem and the top result wa something aboout the pidgeonhole principle?

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27956396)

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27957679)

and, obviously

so what's the problem called?

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27957718)

the 1st thing u said was that he rephrased the problem so we wouldnt be able to google it

the 2nd thing u said was that nobody on this shitsite would be able to solve it

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27957739)

to be honest u dont sound like a math bro

rephrase you're questions more coherently & i'll be happy 2 answer them

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27957809)

And yes, 1 unit in any direction counts.

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958349)

Put A at 0,0. Theres 8 spots away that 0,0 can touch. Theres no way you can B and C in those spots without any touching eachother

Did i fuck up

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958522)

(Also, because I don't think I made this clear, you're considering all points in the plane, not just those with integer coordinates.)

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958678)

Interesting shit tho I like these questions

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958703)

http://www.cut-the-knot.org/proofs/three_colors.shtml

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958691)

Should I post another problem thread? It won't be Googleable this time.

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958733)

Wanna go 2/2 before i pass out

(http://www.autoadmit.com/thread.php?thread_id=2888104&forum_id=2#27958749)