Saturday, September 15, 2001
M I N D  G A M E S


Inspector Dhurandhar in Pascalís den

"PASCAL in city," Inspector Dhurandhar Bhatawadekar, the best policeman in Mumbai, reads the headline and tears up the newspaper. "How did that twisting little Pascal dare to return." "Sir, I believe that he is here to reestablish his illicit liquor business," says 120, the constable. "Aha! So, it will be my barrel against his barrels again," says Inspector Dhurandhar, looking at the barrel of his gun.

"Sir, this Pascal does not look like our Pascal," says 120, to the inspector after removing his eyes from the binoculars. "Shut up! All Pascals are alike, but this Pascal remains shut in his house all day. I suspect he is hiding something." says the inspector. "You are right Sir! His neighbours say that he is hiding something that no one has been able to find so far. He talks about some triangle all the time," says 120. "That must be the golden triangle; come, let us raid the house," says the inspector.

 


Thatís how the numbers of the Fibonacci series (every digit the sum of previous two) are hidden in Pascalís Triangle. Find at least one more pattern hidden in this triangle and tell me at: adityarishi@kasparovchess.com

In the house: "How could you break my door like that," says the man to the policemen after they have barged in. "I can break your nose as well, or else tell me about the triangle," "Sure, but who are you?" "My name is Inspector Dhurandhar Bhatawadekar and every criminal fears it, now tell me your full name, Pascal." "Blaise Pascal." "From where have you come?" "France," "Whatís this triangle?" "Well, Pascalís Trinagle is something that calculates the probabilities of winning in gambling." "Aha! A gambler! What a catch!" "No, no! The triangle is more a tool for mathematicians than gamblers." "Show me."

Pascal: "Each number in a Pascalís Triangle is the sum of the one above it and the one to the above-left. At the tip of the triangle is the number 1 that makes up the zeroth row. The first row contains two 1s, both formed by adding the two numbers to the left and the right above these, and so on. In this way, the triangle is bordered by all 1s and the rows of the triangle are infinite."

"The sum of the numbers in any row is equal to 2 to the nth power or 2^n, when n is the number of the row.

2^0=1

2^1=1+1=2

2^2=1+2+1=4

2^3=1+3+3+1=8

2^4=1+4+6+4+1=16."

"If the 1st element in a row is a prime number, all the numbers in that row (excluding the 1s) are divisible by it. For example, in row 7 (1, 7, 21, 35, 35, 21, 7, 1), 7, 21, and 35 are all divisible by 7. If a diagonal of numbers of any length is selected, starting at any of the 1s bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. This is called the hockey-stick pattern. 1+6+21+56=84; 1+7+28+84+210+462+924=1716; 1+12=13."

Inspector Dhurandhar: "Are you the same Pascal who is mentioned in school textbooks?" Pascal: "I am; so, now you see." "Yeah, I see well. I wanted to arrest you since my childhood for writing that mind-bending theorem. Arrest him 120."

ó Aditya Rishi