Yesterday Professor Pheno Menon took me and one of my friends to his house to demonstrate one of his most recent mathematical discoveries. When we got there, he showed us a rectangular table. He gave us a packet containing quite a few coins and said:
"Friends, see that table over there? Last night I tried almost all possible configurations and came to my phenomenal conclusion: this table cannot be completely covered with $400$ coins like these."
I was well aware of the phenomenal mistakes he often makes; besides, the tables seemed too small for such a bold statement. Seeing disbelief in my eyes, he went on:
"Don't think it's true? OK, go ahead, here are your $400$ coins. These are all yours if you can cover this table with these."
Now I closely noticed the coins. They were all identical and perfectly circular disks, with a very small thickness. So I immediately got to work. After half an hour of effort (while my friend and Professor were lazing on the couch); I came up with a configuration.
"Look Professor, I knew you were wrong," I said, "I didn't even need all the coins. I have arranged $100$ coins on the table such that none of them overlap, and it's impossible to put another coin without causing overlap."
"No no, my boy, you've got me wrong." he replied. "You need to cover the table entirely; there must not be gaps. Every point on the table must lie below some coin. Your coins may, however, overlap."
My friend finally spoke out, "Then, Professor, you're certainly wrong."
How did my friend know?
- For a coin to lie on the table, its center must lie on the table.
- This is not lateral thinking; you must supply a valid mathematical proof that $400$ coins are enough to cover the table using the given information.
- This was recently posted (with a different wording) on another online forum by a user who claimed to have heard this from his friend. The exact source is unknown to me.