Question

➲ ǫᴜᴇsᴛɪᴏɴ:⍰

ᴛʜᴇʀᴇ ᴀʀᴇ ɪɴғɪɴɪᴛᴇ ʙʟᴀᴄᴋ ᴀɴᴅ ᴡʜɪᴛᴇ ᴅᴏᴛs ᴏɴ ᴀ ᴘʟᴀɴᴇ. ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ᴛʜᴇ ᴅɪsᴛᴀɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴏɴᴇ ʙʟᴀᴄᴋ ᴅᴏᴛ ᴀɴᴅ ᴏɴᴇ ᴡʜɪᴛᴇ ᴅᴏᴛ ɪs ᴏɴᴇ ᴜɴɪᴛ.✌️​​

Answer 1

Answer:

Well it says one unit

Step-by-step explanation:

one unit can be anything like cm or inch or in this case zero so it is like 1x0 which is 0 because the whole plane is covered with dots (meaning no space in between)

Answer 2

Answer:

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ᴛʜᴇʀᴇ ᴀʀᴇ ɪɴғɪɴɪᴛᴇ ʙʟᴀᴄᴋ ᴀɴᴅ ᴡʜɪᴛᴇ ᴅᴏᴛs ᴏɴ ᴀ ᴘʟᴀɴᴇ. ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ᴛʜᴇ ᴅɪsᴛᴀɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴏɴᴇ ʙʟᴀᴄᴋ ᴅᴏᴛ ᴀɴᴅ ᴏɴᴇ ᴡʜɪᴛᴇ ᴅᴏᴛ ɪs ᴏɴᴇ ᴜɴɪᴛ.

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Assuming every point of the plane is either white or black, here is a quick "constructive" way to find two points of opposite colour at distance 1. Since there are both white and black points, the infimum r of the distances between white and black points is well defined. If r>0 then there exist a black-white pair at distance d with r≤d<2r, and the midpoint between them is at distance d/2<r of either, so it cannot be black or white by the choice of r, a contradiction. So r=0, and there exists a black-white pair at distance d<1 of each other. The circles of radius 1 centered at these points intersect, and pairing the two centers with the two intersection points one gets at least one black-white pair at distance 1 (in fact one gets two pairs).