Monday, September 27, 2021

Problematics: Cricket with three arms and nine fingers

In this special episode written mostly by Problematics readers, a look at an alternative universe

(Left) C R Sasikumar’s illustration of a bowler on Planet 3.9; (right) a possible numeral system. (Source: Express illustrations)

The cricketer in the illustration above began as a figment of my imagination, after which you, the readers of this blog, helped him develop his traits, characteristics and personality. For this special episode of Problematics, I invited a group of dedicated readers to explore life in an alternative universe, particularly its cricket and its mathematics.

The concept of a century is at the root of the discussion, though it branches out to other things. Cricketers celebrate a century because it’s three digits. But must three digits always begin at the number we know as 100? Let’s find out.

Idea#18.1: Our ancestors decided on single digits for 1 to 9, followed by two digits from the next number. Why choose 9 + 1 = 10?

Dear Kabir, the human race must be following the base 10 for the simple reason that the fingers would have been the first and most convenient counting aid used. It is expected that the numerical system our ancestors developed would be in line with their counting practices.

Jaysun Antony Alumkal (Coordinator, Academic Committee, IIM Raipur)

Idea#18.2: Imagine Planet 3.9, whose inhabitants have three arms, each hand with three fingers. What we know as 8 + 1 = 9, therefore, is 8 + 1 = 10 out there. So how much is a century?

Dear Kabir, with base 9, a century on Planet 3.9 is 9 × 9 = 81 in human terms. And 100 on Earth = 121 on Planet 3.9.

Biren Parmar (PhD student, Texas A&M University)

Idea#18.3: Sachin Tendulkar had 51 Test centuries and 35 scores between 80 and 99, including one of exactly 80. If an author on Planet 3.9 writes a book called ‘Sachin, the Cricketer of Earth’, how many centuries will she count?

Dear Mr Kabir, there are 85 scores of 81+. Since 81 is a century on Planet 3.9, we can say in Earth terms that there are 85 centuries there. So, 81 + 4 = 100 + 4 = 104 centuries on Planet 3.9.

Sampath Kumar V (IIM Kozhikode alumnus)

Idea#18.4: For a cricketer, how useful are three arms?

Dear Kabir, I would prefer a longer third arm, and why not multiple joints. That way the middle arm can be extended to full length or curled (jointed) up to a third or fourth of the other arms. The middle arm can be used like the kangaroo uses its tail. It can be used for reaching longer distances or curled up and kept out of the way. Imagine a bowler running in and using the middle arm to thrust upwards at the time of delivery. What a bouncer it would be, delivered from a height. Or the batsman using his third arm to leap in the air to counter the bouncing ball. The fielder with the third arm extended could take far-reaching catches.

Sathya Prakash (Software developer, New Jersey)

Idea#18.5: A half-century on Earth is how much on Planet 3.9?  And vice versa?

Dear Kabir, 50 in base 10 = 55 in base 9.

M Natrajan (IIM Kolkata, batch of 2009)

***

Dear Mr Firaque, assuming that 10 × 10 is a century for a human (fingers raised to the power hands), 9 × 9 × 9 would be a century for a Planet 3.9 inhabitant. So, a century on Planet 3.9 is 729 to humans. Assuming that our half is half because we have two hands, their half will be one-third because they have three hands. So, possibly, their half-century is 729/3, or 243.

***

Dear Mr Kabir, going by the reader’s interesting argument that “half is half because there are two hands”, then there would be two partial-centuries. In earth, we have one partial-century called half-century, likewise they will have two partial-centuries called one-third century and two-thirds century. So, 27 and 54 will be those two partial-century landmarks.

Sampath Kumar V (IIM Kozhikode alumnus)

Idea#18.6: A theory for how humans invented the concept of odds and evens. If a hunter on Earth caught 2 rabbits and picked up 1 in each hand (equal distribution) it was even. If he caught 3 rabbits and distributed them 2 and 1 (unequal distribution) it was odd. So what are odd and even on Planet 3.9?

Dear Kabir, in Planet 3.9, multiples of 3 would be even and the rest would be odd. I think it would also affect the understanding of left vs right, north vs south, etc. I think they would always think of three ways instead of a binary yes and no. They could have a tri-state which is probably between yes and no. How about a “don’t care” state which is basically indifference?

M Natrajan (Class of 2009, IIM Calcutta)

Idea#18.7: Beyond cricket, are three arms a help or a hindrance?

Dear Mr Kabir, having often wished for an extra pair of arms, I don’t believe they’d be a hindrance at all if there was sufficient brainpower to direct each of those arms separately.

Anindita Basu (IBM India)

***

Sampath Kumar V adds: “In writing and drawing, there will not be much difference. We are used to writing with one hand only. In computers, imagine the speed using three hands simultaneously. With three arms, one could carry more luggage or shopping bags, hold three children at one time while crossing the road, use two hands to support while aiming in shooting.”

Idea#18.8: Are three fingers a help or a hindrance?

Dear Kabir, five fingers in humans might seem indispensable today, but I’m sure if it were a different number, we would have figured out ways to live with that. Similarly, three fingers are neither an advantage or disadvantage for 3.9’ians, assuming we are working with intelligent beings.

Biren Parmar (Texas A&M University)

***

Dear Kabir, I have rabbits at my house and I guess I can hold two of them (smaller ones) by their ears with three of my fingers. Assuming par-human intelligence, they will design their modern life exactly suited for their anatomy. Kabir, the discussion was an entertaining one.

Jaysun Antony Alumkal (IIM Raipur)

***

Dear Kabir, I did try doing a few things with just three fingers, e.g. holding the steering wheel, writing, holding hands with my wife, holding a fork and knife, and was pretty successful in all of them.

Sathya Prakash (New Jersey)

Next week, the solutions to last week’s puzzles, and of course a new set of puzzles. Meanwhile, can we have lots of feedback on this special episode, please?