Saturday,  August 11, 2001
 M I N D  G A M E S
 The Judgement Day THIS is a vast territory, stretching from zero to infinity, where time and space are curved and hold no meaning anymore (The world as we know it has ceased to exist): A 40-mile-long army column advances 40 miles as Sarah Connors rides on horseback from the rear of the column to the front and back to the rear. She travels at a constant speed, either parallel or opposite to the movement of the army, searching for the terminator among the chaos in the column. As Alan Lindsay would have put it: like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem. When the army ends its journey, the terminator will destroy whatever has been left of the world. Sarah’s thinking: "It was a fine day when it began and, now, it is a race against time to save the world. God, please tell me where should I look for some help." "In the column," she hears a voice in her mind. "God! Is that you?" she says in a trembling voice. "Yes, my child," says God, "The terminator will ask you a riddle at the end of the journey and destroy the world if you fail to solve it. However, as the time has curved, you will find the best mathematicians of all times, who can help you, in the column."

 On earth there is nothing great but man; in man there is nothing great but mind. — Sir William Rowan Hamilton It has been a year since we started playing Mind Games and the judgment day has arrived. It is difficult to adjust infinity in a finite space and, a year ago, I was told that it would be hard to sustain this column. However, mathematics never ends, for the end of it would be the end of the world, and the journey continues. Have you enjoyed this journey? Write to me at ‘Mind Games’, Windows, The Tribune, or adityarishi@kasparovhess.com

Sarah turns her gaze back to the column and finds a man, in 1801 AD, measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken. "Who are you and why are you studying the mountains?" she says. "I am Karl Friedrich Gauss, looking for evidence that the geometry of space is non-Euclidean." Sarah says, "I need your help, please stay close." Next, she sees a man who is making geometrical figures on paper. He calls out to her, "Look what I found! If we construct equilateral triangles on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles also form an equilateral triangle. By the way, I am Napoleon Bonaparte." "Stay close, we might need you," says Sarah. "ABLE WAS I ERE I SAW ELBA," replies the emperor with his favourite palindrome.

Sarah’s eyes rest on Galileo in 1610, while he is writing a coded message for Kepler: smaismrmilmepoetaleumibunenugttauiras. Kepler decodes it as Salve umbistineum geminatum Martia proles (Be greeted, double knob, children of Mars.) He deduces that Galileo has discovered two moons of Mars. Galileo, however had written Altissimum planetam tergeminum observavi (I have observed the highest of the planets — Saturn — three-formed). Sarah calls them, "Hey! You two, do you want to save the world?" as they look up.

The journey ends and terminator appears with the question: "Sarah, How far have you ridden?" "Forty miles to the front and 40 miles back, that makes it 80 miles, but hold it for a second. I let my friends give you the answer," says Sarah and signals her army to step out of the column. The mathematicians say, "Let A = speed of the army, H = speed of the horse, and d = distance traveled by horse. The time for the army to march 40 miles is 40/A. This must be equal to the time for the horse to get to the front = 40/(H-A), plus the time for the horse to return to the rear = 40/(H+A). The equation reduces to 40/(H-A) + 40/(H+A) = 40/A, i.e., the quadratic H^2 -2*H*A -A^2 =0, i.e., H=A*(1+sqrt(2)). Therefore, D = H * [40/A] = [A*(1+sqrt(2))]*40/A = 40 * (1+sqrt(2)=96.569 miles." "You have saved the world," the terminator delivers the judgment.