# Interesting Math

1. Interesting Math
111,111,111 x 111,111,111 = 12,345,678,987,654,321

2. Originally Posted by AffiliateHound
111,111,111 x 111,111,111 = 12,345,678,987,654,321
Someone has entirely too much time on their hands.

Someone has entirely too much time on their hands.
That would be the person who sent it to me.

4. It's actually pretty easy to visualize how it works that way if you write out the long multiplication. A smaller example:

Code:
```   1111
x 1111
-------
1111
1111
1111
1111
-------
1234321```

5. Amazing! I think it doesn't need a calculator?

6. It's actually pretty easy to visualize how it works that way if you write out the long multiplication

7. Nice! How about this one: You can cover a 64 square checker board with 32 dominoes. Each domino takes up 2 squares. However, if you remove the top left and bottom right squares from the board, it is impossible to cover the 62 squares left with 31 dominoes. Why?

Nice! How about this one: You can cover a 64 square checker board with 32 dominoes. Each domino takes up 2 squares. However, if you remove the top left and bottom right squares from the board, it is impossible to cover the 62 squares left with 31 dominoes. Why?

No matter where you start, after 30 dominoes, the last two open squares do not touch.

9. Originally Posted by AffiliateHound
No matter where you start, after 30 dominoes, the last two open squares do not touch.
Can you prove that for every single combination of 31 dominoes? One way to do this would be by "exhaustion", listing every single possibility for the first 30 dominoes. There would be 62 places for one leftover spot and 61 places for the other, resulting in 3782 ways, not including all the ways to place the 30.

There is a fairly simple proof for this however. You seem to be very close to getting it, based on your statement that "the last two squares do not touch".

10. Every domino will cover both a black square and a red square. Because two red squares are removed (leaving just 30), it's impossible to put more than 30 dominos on the board.

11. Originally Posted by MichaelColey
Every domino will cover both a black square and a red square. Because two red squares are removed (leaving just 30), it's impossible to put more than 30 dominos on the board.
Yes, that is it!!