Combinatorics - 3

Combinatorics - 3

PREVIOUS

Kabani was making progress with her counting of how many possibilities are there in anything she did. Sometimes she found it easy like switching a light on or off, sometimes it became tougher like the possible lengths of a line she could draw with a 15cm scale. Her parents couldn't help her either on this. Her unsolved problem list was ever-increasing now. It was a big motivation to explore further...

Kabani observed that while calculating the number of possibilities sometimes she was adding and sometimes she was multiplying and she never did them together. Now this was strange! She wasn't told where to do what, then how did she know - when and what to do. Look at the following problems -

There are 5 apples and 3 oranges, the number of ways you can select an apple or an orange is ______.

There are 5 apples and 3 oranges, the number of ways you can select an apple and an orange is ______.

The two questions varied in just one word and that makes all the difference. When Kabani was given the choice to select an apple or an orange, she was selecting one object out of the 8 objects as there was no emphasis on what she should be selecting. So the number of ways she could have done that was 8. However, when she was told she had to select both an apple and an orange, the situation is very different. She can select any one of the 5 apples, so the selection is possible in 5 ways. Once she has selected the apple, she can select an orange in 3 ways. After every choice of an apple there are 3 further choices. So in all she can make a selection in 3 + 3 + 3 + 3 + 3 (=15) ways, i.e., 3*5 or 5*3 ways. (Do you remember the law which says 3*5 = 5*3?). What Kabani thought was multiplication was just repeated addition! But why repeated addition?

The reason lies in the fact that, in the first case Kabani was doing only one work but in the later she had two tasks. She could do the first in a few different ways and then later do the second in few more ways, thereby resulting in a huge number of ways she can complete both the tasks together.

Mathematicians call the first case as the law of sums and the second as the law of products. The names hardly matter... may be they should better be called as the law of addition and the law of repeated addition. It is for you to decide, what you would like to call them. How about Alpha Rule and Beta Rule? You are now one step ahead in your understanding of combinatorics, that is what matters. 

Until next time -

A "sign" is being assigned to every person in a village. Each "sign" consists of a geometrical figure (triangle, square, rectangle or circle), an alphabet and a single-digit number. How many unique signs can be made?

NEXT (coming up)

Dance for navigation

"The bees are buzzin' in the tree, to make some honey just for me…….If you act like that bee acts, uh uh you’re working too hard"- Baloo in the Jungle book

Bees are definitely one of the hardest workers on the planet, with ants being their only close competitors. Bees are social insects, having a hierarchical family group. At the top is the queen and the bottom most are workers who bring in honey to feed the ever growing family. Drones, whose only job is to fertilize the eggs that the queen lays (she actually lays a lot), and larvae in various stages of maturity are the other occupants of the beehive. But why do bees buzz and wiggle around so much, wasting their energy, as if the energy spent on finding and bringing honey wasn’t tiring enough? Well if you have keenly observed bees hard at work, you would realize that all that wiggling and buzzing is their way of communicating with each other. Yes they dance and buzz to convey to their fellow bees the location of a food source In case you never got to observe bees, then let’s do a bit of eavesdropping and find what they do. Make no noise! You don’t want to anger them and get stung.

Get down on the Dance floor



So lets follow the scout bee who is just entering the hive, he seems to be really happy and dancing in a particular way. Well the dance floor is usually near the entrance, but in winters it may move more inside the hive and in real hot climate they use the area outside the hive as dance floor. Since natural hives hang vertically, the dance is also on vertical plane and the bee has to tell a route that’s horizontal, that requires lots of sense of direction and distance (I am already wondering which plane is horizontal and which is vertical). When the dance is outside the hive (or in artificially constructed honey combs), it is generally done on a horizontal plane.

The Round dance

The bee that just entered is making vertical circles. Hey!!! See that he has suddenly reversed direction and is making a circle again. There are so many rapt observers (other bees I mean) besides us.

Let me translate it for you. Circles mean the food is nearby not more then 100 meters away. Having received the good news, the other bees, all fly away in all directions and search within 100 metres radius (hard work), but an experienced bee, using the odour given out by our scout, directly flies to the flower referred to by the scout’s odour (smart work).

The tail-waggle dance

Ohh, we have another arrival! The poor creature looks so tired, but excited nevertheless. So he is taking to the dance floor as well, meaning he too has found food. His moves are different though. Look! He is making a straight run and wagging and buzzing rhythmically (thought to be describing food quality and quantity);turning to left, making a semicircle, back to where he started; he is doing the straight run again, but look, now he is turning right and making a semicircle; making a near 8 figure each time. Some of them have joined in his dance. And the spectators have taken off, so sure of where they are headed. However strange this looks to you, it’s the most ingenious way of conveying the direction and the distance of the food (considering that they can’t speak as we do).
Well, tail waggling dance is performed when food is more than 100 meters away. It uses the angle of sunlight (or partially polarized light*- implies they can convey direction even when the sun isn’t directly visible, say as on a cloudy day or when sun is setting behind a mountain) to describe the direction and number of eights to describe the distance. Bees must be having the most accurate sense of time (since sun and its rays change position as the day progresses; also consider that he has to correct for its position from the time he found food to the time that has lapsed till his dance performance).With increase in distance the number of 8s per unit time decreases and the length and duration increases. So if for a food source 100 meters away the bee makes 10 eights in 15 seconds, for 3km it will make one stately 8 in the same time. Waggle part or the straight run is what conveys the direction. A straight vertical run upwards means the food is towards the direction of sun, downwards implies away from sun. If it is say 30 degrees to sun rays angle, then the run will be at an angle of 30 degrees to the vertical.


Courtesy: www.answersingenesis.org, www.theevidence.org

On days when the dance floor is shifted to the outside the sun itself may be used as reference. Sometimes even the onlookers join in the dance, learning the moves they are actually by-hearting the location.
This language of the bees was first deciphered and translated for us by Karl von Frisch when World War II was on in full swing (when people were decoding other stuff). He understood the bee language so well that he could tell his neighbors the exact distance and location of a flower from which the bees in his side were feasting upon. That’s some marvelous decoding Mr. Frisch! In fact he made artificial honey combs, exposed the bees to artificial polarized light and studied their dances out and out. But that’s not all to it, the dialect and the accent of the bees varies with regions. For e.g. An Austrian bee (von Frisch worked on these) does round dance for a distance up to 100meters, but an Italian bee shifts to tail-waggle after 80meters. So if you made an Italian honeybee talk to an Austrian one, you would most probably end up confusing the Austrian. The bee language is no less complicated than ours but nevertheless well structured to be precise every single time as if a GPS was inside their head.
* Tie a rope to a pole at one end, stretch it to its length and pluck it. It forms ripples in only one direction, the direction along which it was plucked. Similarly polarized light waves also vibrate only in one direction (light waves are polarized by polarizer and also by scattering). Sunrays get scattered by molecules in atmosphere and become partially polarized. Although the amount is going to vary with region and the sun’s location in the sky, the pattern is very much predictable. And it’s these patterns that the bees recognize and use for their communication.

Reference:

Bionics,Vincent Marteka;Lippincot, 1965

Decode the bar code

Ever wondered what differentiates a can of diet coke from a jar of peanut butter? If yes then welcome to the club. For answer, one has to look into the ubiquitous black and white stripes on these products, the barcode.

History of barcode dates back to about six decades, when it was invented by Bernard Silver in 1948. (Twenty five years gone by, before it was first used to read a price on a juicy fruit gum on June 26th 1974).Today it act as both a mundane fingerprint of modern life and a cultural icon of retail shops across the world.

Anatomy of a Bar Code is composed of a series of parallel black and white lines. When a scanner's laser beam hits the bar code, the black modules absorb the light, while the white modules reflect light. A photo diode in the scanner decodes the reflected light into an electrical signal, which is amplified and translated into digital code by the scanner's microprocessor (Fig1). That code is then fed into the store's main computer, which finds the price.



Fig 1

Different bar-code languages are used to identify different types of products, like groceries, clothing and electronics. Although many bar-code languages have been developed by the retail and manufacturing industries since they were first adopted in the early 1970's.The most widely used type is the Universal Product Code (U.P.C), found in most groceries and retail stores.
The U.P.C. is a 12-digit code: the first digit identifies the general category of the product; the next five, the product's manufacturer; the next five, the individual item (like Coke, Diet Coke or Caffeine Free Coke), and finally a ''check digit'' that is used to make sure that the code is scanned correctly in the right orientation.

A typical U.P.C., read from left to right (Fig 2), starts with a quiet zone Next comes the left guard pattern, which alerts the scanner and computer system that information is coming. The next modules identify the number of the product category. That number is printed at the lower left margin of the U.P.C. manufacturer. The next five digits identify the manufacturer. Centre guard pattern divides the left and right halves of the bar code. The next five digits on the right side identify the particular product -- e.g., whether it is a 12-ounce jar of Brand X crispy peanut butter or an 8-ounce jar of a smooth one. The check digit (the last digit on the right) is used to make sure that a U.P.C. has been scanned correctly. The computer does the following calculations; if it comes out with the check digit, the scanning was correct. For a U.P.C. that starts with 0 12345 67890: Add the digits in odd positions: 0 + 2 + 4 + 6 + 8 + 0 = 20 Multiply the results by 3: 20 X 3 = 60 Add the digits in even positions: 1 + 3 + 5 + 7 + 9 = 25 Add the last two results: 60 + 25 = 85 Subtracting that answer from the next-highest multiple of 10 should produce the Check Digit: 90 - 85 = 5.




Fig 2

Next time, when u go to a supermarket figure out what these lines implies. Now you know what it means!