Thought Provoking Puzzle

I think we should start from a smaller board and then work our way up to try to find patterns.

My hypothesis is that the best we can do it to pick all the points on the two outside edges.

Is using R cheating?

For 635,376 valid positions, there are 784 rectangles which means there are 634,592 maximum number of points that are possible.

And I know... I know... apply instead of for loops.

Alright. Here is the answer.

this, right?

New Question

I don't know the answer to the OP's question without thinking some more. But the nature of the puzzle is slightly like a geometry problem which was Ted Kaczynski's last published piece of professional mathematical work.

Perhaps Pick's theorem could be used in some way, as it was used to solve Kaczynski's puzzler. Or maybe not - I'll think a bit.

en.wikipedia.org/wiki/Pick's_theorem

You can move the red dot to literally any other position (60 of them) and it would be four selected points that don't form a rectangle. I don't understand the 24 answer.

>statlet """"coding""""

The problem is equivalent to coloring the edges of a graph on n vertices with n colors such that any vertices connected to the same color are maximally connected with that color. Then color the graph in such a way as to maximize the number of vertices touched by each color. A complete graph on 8 vertices can be colored into 8 triangles, yielding 24 points. Each color represents a row on the nxn matrix, and each vertex represents a column. In other words, each row of the 8 by 8 matrix should have three dots selected such that each row only shares at most one dot with each of the other rows.

...

ment for

fuck off, nobody cares about discrete """""""""""""""""math"""""""""""""""""

I see some rectangles and squares here.

maybe you dont like discrete math, but your big faggoty mouth could benefit form taking on a more discrete approach

boom, wha now stoopid bich

then stop looking in da mirror fegget

boom

I see 3 valid points of your example.

Do you know what a rectangle is?

Or do you choose a narrow definition to make the problem easier?

Do you know what "parallel to the edges of the grid" means?

Or did you not read the OP all the way to the end?

>Do you know what a rectangle is?
actually at first i thought they said rectangle because they ment equilateral rectangles were allowed, and it made it way harder
warosu.org/sci/thread/S9168687#p9170507

either way, i didn't solve it by math. get fuxed mathtards

I have a way of getting 24=8+7+5+3+1
Not sure how to prove that's the max, though.
Move outer rectangles inside.

...

There are (n^2-n)/2 possible pairs per row. If there are k points in a column then that column takes up (k^2-k)/2 of those possible pairs. If the number of points per column is held constant, this gives us the upper bound

n(k^2-k)/2 =< (n^2-n)/2

k^2-k =< n-1

k =< (1+sqrt(4n-3))/2

So for n = 8, k =< 3.19...

This gives us 8*3 points, 24.

If the number of points per column is not constant then we have

sum(k^2-k)/2 =< (n^2-n)/2

sum(k^2-k) =< n^2-n

and we want to maximize sum(k)

Now why isn't 4 points in one row and 3 points in every other row the solution and 28 column pairs, the maximum possible in an 8x8 grid? This would give us 25 points, one more than 24.

Well if we take 4 points in one row that leaves us with only 4 columns that do not have all of their pairs taken (6 pairs). To take three points in another row we need at least one of those column pairs, but there are only 6 of them and we have 7 rows left. So 25 is impossible, making 24 the maximum.