# Robots on a Plane

A proof that, should you ever be attacked by a giant robotic bishop while on a chessboard, you can survive by standing in the right spot.

at any time is even.

# Basis

.

Thus, is true.

# Inductive Step

Assume and is true, we show that is true.

The coordinates resemble the location of the robot at time .

Because is true, is even.

Thus, because is true.

At time there are three cases:

## Case 1: The robot moves up and to the right from

Therefore, coordinates are

thus,

by substitution

where

Therefore, is even.

## Case 2: The robot moves either left and up or right and down from

Therefore, coordinates are either or .

Thus, or

Therefore, is even.

## Case 3: The robot moves left and down from

Therefore, coordinates are

Thus,

where

Therefore, is even.

# Conclusion

Therefore, .

Thus, no matter which way the robot moves, will always be even.

And thus, no matter how much time the robot has to move around, it will never be able to reach the point , since is odd.