Star Trek: Odd victory

If the number of odd numbers in any sequence is odd, putting a plus or a minus sign between successive numbers and computing the result afterwards will always give you an odd number.

CAPTAIN'S log, star date 2345: These are the voyagers of starship Enterprise, on its five-year mission to explore strange new worlds, to seek new civilizations and new lifeforms, to boldly to where no mathematician has gone before.

The voice of Captain James T. Kirk: "Something strange is happening to men on Enterprise. The trouble seems to have begun after the arrival of a computer on this ship. This tin box should not be judged by its size, it is a highly intelligent combination of wires and metal created on planet Gama 3. The human mind is too weak against this machine that is slowly taking away the control of Enterprise from me. However, Mr Spock looks unaffected by it. He has been in the same chamber with the machine for the past two days."

The ship’s doctor, Dr McCoy, enters the chamber. Kirk says, "Bones, you don’t look good, what happened now?" Mr Spock has also entered the room, unnoticed by the doctor. McCoy: "The machine is a devil’s instrument Jim, it has just killed your navigator. This is the 15th person who has been killed since the damn thing arrived on board. The chief of Gama 3 station said it was the finest computer ever created on his planet and it never made a mistake."

"That is true doctor," says Spock, "but the computer was also programmed to eliminate all things that made any error. It killed the navigator because he could not find out the shortest route to planet Triton 4." McCoy: "That’s devilish! Peter was our finest navigator." Spock: "Finest, but not perfect, the computer does not accept anything but perfection from men." Kirk: "We are humans, Spock, how can we be perfect? Assumption and estimation is the key to all logical conclusions for us, imagination is our lifeblood."

Spock: "Unfortunately captian, the computer does not understand that." Kirk: "Can we destroy it?" Spock: "Negative captian, it is too fast for our weapons." Kirk (loudly): "Hoover!" Spock: "Why did you say that." Kirk: "Because it is the world’s biggest dam." Suddenly, there is a smile on Kirk’s face. Kirk: "Spock, will the machine destroy itself if we prove to it that it has made an error?" Spock raises one eyebrow and says, "It might be possible, captain." "Jim!" says Bones and embraces Kirk.

The three officers enter the bridge of the ship where the computer has moved now. "Computer, we have a task for you, can you do it?" says Jim. An animated voice replies, "I can do any task." Kirk: "Good, I have a simple game for you where you begin with a sequence of numbers. Taking turns, you and I place either a plus or a minus sign between any two successive numbers. When we have filled all the empty spaces between the numbers, the arithmetic result will be calculated. You win if the result is odd. If the result is even, I am the winner." Computer: "I must warn you, if you lose, you will be destroyed." Kirk: "I had assumed as much."

Kirk and the computer stand in front of the computer screen. At the press of a button, a + or a - sign is inserted between two successive numbers. The selection of a sequence is, however, not random. Captain Kirk chooses the sequence everytime because he is the challenger.

Whatever, sequence Kirk writes, the number of odd numbers in it is always even. Computer tries the game several times, but loses every time. Kirk: "Computer, you have lost." Computer: "I have lost." Kirk: "You have made an error, now destroy yourself." Computer: "I have made an error. I must kill myself." Lights in the machine go off and sparks fly off it.

"Jim, how did you do that?" says Bones. Kirk: "The result only depends on the number of odd numbers in the sequence. If this is odd, the result will be always odd, regardless of the distribution of signs. To see this, pick up a number A. Add and subtract from it another number B, so that C1=A-B and C2=A+B. Then, since C2-C1=2B, C1 and C2 are always of the same parity, i.e., either both are odd or both are even. Thus, the selection of signs does not affect the parity of the result. The latter will be odd if and only if the sequence contains an odd number of odd members." Another adventure next week.

— Aditya Rishi