When you're viewing this video don't cheat and look at the answer below. Don't blame me if you get your computer screen all smudged!
What The Heck is Boolean Algebra?
A computer 'thinks' in terms of ones and zeroes (known as a bit). This is what's known as boolean algebra. In a way, it's even simpler than the decimal system as you only have two numbers to worry about. In the above puzzle, once you figure out the template and the pattern the solution becomes almost child-like in its simplicity.
Take A Look at the Number Pad on Your Keyboard
It probably looks like this:
7 8 9
4 5 6
1 2 3
We notice that there are four even numbers and five odd numbers. If we replace the letter O for all the odd numbers and E for the even numbers we end up with this:
O E O
E O E
O E O
So long as the numbers one through nine are placed in an orderly fashion the template cannot fail.
Three Rules of Addition
In the following 'odd' is any odd number and 'even' is any even number.
Rule 1: odd + odd = even
Rule 2: even + odd = odd
Though this doesn't apply to our riddle,
Rule 3: even + even = even
And Your Point Is?
When we start on an O square and given an odd number of moves, we will always land on an E square, no matter what (Rule 1). So our pal Richard can remove any O square he wishes. As we stand on an E square, we are again told to move an odd number of moves. Naturally, we end up on an O square (Rule 2) and Richard can confidently remove any E square he sees fit.
The pattern of alternately removing O squares and E squares continues until we run out. As there is a 5-4 imbalance of odds and evens, an O square is the first to be removed (top right) and must be the 'last man standing' (bottom right).
So What Threw Me Off?
Puzzles like these are notorious for their red herrings and dead ends. The symbols are arbritrary as is the order in which they are removed. So long as an O square (or an E square) is removed when it should be, the pattern holds according to the template. So long as we are consistently given an odd number of moves it doesn't matter.
Hope you enjoyed that as much as I did!