Riddle #7: Midpoints
Last weeks riddle was solved by @themadrunnah, congrats to him for claiming the SBD prize!
This weeks riddle goes as follows:
How can you place 5 points on a plane such that:
- All of the points have integer coordinates in both x and y (like (1, 4) or (6,2))
- None of the midpoints of line segments formed by connecting any two of the 5 points have integer coordinates in both x and y
A valid solution must either list the coordinates of the 5 points or prove that 5 points satisfying the above do not exist.
Prize
This week, the first person to upvote and comment with a valid solution (as decided by me and only me) prior to this post paying out will win 1SBD! The prize will be paid out right after the rewards from the post are paid.
Don't forget to follow and to check out my past riddles
First off, thanks to everyone for all the support! If you'd like to browse my older riddles you can do so at the links below:
https://steemit.com/life/@droopy/steem-riddle-of-the-week-2-flipping-cards-prize-doubled
https://steemit.com/life/@droopy/steem-riddle-of-the-week-3-alternating-truth-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-4-black-cards-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-5-the-cup-game-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-6-burning-ropes-prize-doubled
Your second requirement is not totally clear. For example, is it okay for the midpoint to have one integer value and one non-integer value, like (4,4.5)? or do both need to be non-integer? Just looking at each line you can tell that at least x OR y in every pair is non-integer. If both need to be non-integer, I'll need to keep working.
Good question, either x, or y being non-integer is sufficient for the midpoints to be valid so (4, 4.5) would be fine.
I'm not sure if this is right, it's been a while but it's not possible for a fifth coordinate to exist by the rules because only four coordinates can be plotted with numbers being (odd,odd) (even,even) (odd,even) and (even,odd) and not create an integer coordinate. A fifth one would always create a full integer coordinate plot because there's only odd and even sets of numbers and finding the midpoint of any two coordinates involves dividing by two. Further, since there can only be four combinations of odd and even, one set combination would be repeated resulting in two integers. I don't remember if there's an exception so this'll be my final answer until I see otherwise.
This is correct! I'll ship you the price within 24 hours of when this post pays out :)
Sweet! Thank you very much!
Dirichlet Principle is what it's called?
Amazing
My brain hurts, now I know why I'm creative and draw pictures. I'll take anotjer stab at this puzzle though.... 🤔 hmmm
its definitely a tough one :)
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