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Showing posts from January, 2017

The "Bhakshali" Algorithm

This is a write up to describe an algorithm described in an ancient Indian manuscript. Its called the Bhakshali manuscript. The manuscript describes some mathematical assertions, methods and algorithms that has been dated to several thousands years ago.

A really cool algorithm described in that manuscript is an approximation for finding the square root of a number. What I liked about this algorithm is that its handy. You could quickly approximate the square root of a real number with just some basic division and addition.

This is how the algorithm works:

If 'X' is the number you want to find a square root of, find the nearest whole number 'N' that approximates it. So if X = 23.2 then N = 5. Find the difference between X and N*N. Call it D. In this case it works out to -1.8. This should be too tedious to work out either.Now comes the magical part, divide this difference (D) by 2*N. So that's -1.8/10. Again, this shouldn't be that difficult to do in your head, -0.…

The Bacteria Division Puzzle

Q. A jar has a single cell of a bacteria. After every fixed interval of time the bacteria either splits into two with probability 2/5, does nothing with probability 2/5 or dies out with probability 1/5. What is the probability that the bacteria would continue to make a large family tree over an extended period of time?

A. The situation can be described by the following visual.
Assume that the required probability is 'p'. The term 1 - p would represent the probability that the ecosystem eventually dies out. Each of the above scenarios contributes a quantum of probability towards the ecosystem eventually dying out. Lets start off by represent 1 - p as 'x'. The probability that the bacteria die out is

The total of each of the above must add up to the probability that the bacteria eventually die out, which is 'x'. So you can phrase the problem recursively as

This simplifies to

which is a quadratic equation, yielding a solution as

This further yields x = 1/2 or x =…

The Forgotten Geometric Mean.

Often times a lot of people working with data are trying to create an index of some sort. Something that captures a set of key business metrics. If you are a site (or an app) you want to create some sort of an engagement index, which if trending up implies good things are happening, bad if it is trending down. The creators of such metrics (think analysts) tend to prefer a weighted arithmetic mean of the influencing factors. If the influencing factors are f1,f2, f3 (say) with weights w1, w2, w3 then the index would be computed as

However, what does not get factored in are the final consumers of the index (think product managers) and there could be many. They will invariably try to check it with something else they have handy. For example, if clicks on a site went up 20% the index may be up by just 5% (say) or vice-versa. If resources are being allocated based on the movement of such an index, it will invariably lead to contention on what is the right weighting to be given to each facto…

Colored Cards and Numbers Puzzle

Q: You have a set of thirty six cards. The cards are six in color ( six each) and each color is numbered from 1 to 6. You draw two cards at random. What is probability that they are of a different color and have a different number?

A: The first card can be drawn at random. It does not matter what its color or number is. To compute the probability that the second card is different in color and number from the first, it helps to visualize the situation in a simple way as shown below.

In the figure above, assume the green dot represents the card that was picked. The marked out cards represent the cards that should not be picked to get a different color and number. Also, the act of picking a card bought down the pool of cards from 36 to 35. The remaining unmarked space represents the available set of cards to pick from. This can be computed easily as

This yields an overall probability of

If you are interested in learning the art of probability, some of the best books to learn it from are …