Christopher’s calendar

August 3, 1492: Christopher Columbus is about to leave the shore of Spain in search of India. He is asked to deliver a farewell address to his admirers who have gathered to bid him goodbye. "How will you keep track of the time on your ship?" says a reporter who is "well ahead of his times". Columbus says, "Sir have you ever observed that every fourth year is a leap year unless it’s divisible by 100 and among the years divisible by 100, the ones that are divisible by 400 are still leap." "Yes," says the reporter, but what does it prove?" "Only that calendars are deeply related to mathematics and for doing calculations, you do not have to rely on any gadget. You have your mind for that," Columbus says.

"How strong is your mind honourable captain of the fleet?" says the reporter. Offended by the question, Columbus asks the reporter to select three consecutive dates in a row, column, or diagonal from any month as shown on the calendar. Columbus says, "Write these one beside the another to form a single number. This is your first number. Select another three dates and write these down to get a second number. Multiply the first number by the second number. You may use one of your gadgets that you call calculators. Give me the list of the digits of the product after skipping just one digit. I’ll tell you the digit that you have skipped."

Columbus knows that as far as the properties of monthly calendars are concerned, there is one exception — September 1752. Its table is rather weird. There are no dates from September 3 to 13 in this. It is because of the change from the Julian to the Gregorian calendar that we use today. The new dating system was proclaimed in 1582 by Pope Gregory XIII. The day following October 4 was reckoned as October 15.

Columbus says, "Everything in calendars is based on properties of arithmetic progressions (or series). These are sequences of numbers with a fixed difference between any two consecutive members. In a row, the difference is 1, in a column 7, on diagonals 6 and 8 (or -6 and -8, depending on whether you count downwards or upwards.) Three consecutive terms in any arithmetic series can be written as a, a+d, a+2d. Then the average (a+(a+2d))/2 of the outer two equals (a+d) which is the middle term. Therefore, the sum of three successive numbers is (3a+3d) and is divisible by 3."

"Since the number is divisible by 3, if the sum of its digits is divisible by 3, should we write three dates located successively in a row, column, or diagonal, the resulting number will be always divisible by three. The product of two such numbers will therefore be divisible by 9. Thus, the sum of its digits will also be divisible by 9. So it is easy to detect a missing digit."

"In a 4x4 square, the dates can be written as:

(a) (a)+1 (a)+2 (a)+3

(a+7) (a+7)+1 (a+7)+2 (a+7)+3

(a+14) (a+14)+1 (a+14)+2 (a+14)+3

(a+21) (a+21)+1 (a+21)+2 (a+21)+3

If we choose four dates so that only one is selected per row and only one per column, when we add these up, there will be only one in the form (...)+1, only one in the form (...)+2, and so on. Inside the brackets, we’ll have only one (a) and only one (a+7) and so on. Therefore, the sum will always be equal to 4a+(0+7+14+21)+(0+1+2+3)= 4a+48."

"I thought you would give me the answer now, I am waiting," said the reporter. "My dear Sir, I have just demonstrated how you can use your brain to find that out."

"No person can have a monster memory," says the reporter. Columbus says, "Haven’t you heard of John Wallis who had occupied himself in mentally finding the integral part of the square root of 3x1040 and had written it down from memory several hours later. Mathematician Von Neumann’s ability to do mental arithmetic was legendary. Multiplying two eight-digit numbers in his head was somthing he could do with little effort. Zerah Colburn’s powers in calculating are also almost a folklore in Vermont, USA where he was born in 1804. At the age of eight, he could instantly give the product of two numbers, each a four-digit one. Asked to raise 8 to its 16th power, in a few seconds, he gave the answer — 281,474,976,710,656." "If he was born in 1804, how could you know about it in 1492?" the reporter says. "Time travel, that’s what brought you here in the first place," says Columbus and sets sail.

— Aditya Rishi