# Developing intelligence with the routine preparation

Developing your ability to think seemingly complicated problems is a must if you intend to crack any management entrance exam.

Developing your intelligence, along with the routine preparation for CAT, can be the key to success.

We will begin with a few examples:

Consider the following three questions. It would be ideal if you can begin solving them before you look the solutions.

1) A boy starts adding consecutive natural numbers starting from 1. After some time, he reaches a total of 1000, when he realizes that he has double counted a number. Find the number double counted?
2) A boy starts adding consecutive natural numbers starting from 1. He reaches a total of 575, when he realizes that he has missed a number. What can be said about the number missed?
3) Find the 288th term of the series: ABBCCCDDDDEEEEEFFFFFFG…

Let us spend some time focusing on solutions

Consider the fact that when you add numbers as stated above (1 + 2 + 3 + 4 … ) the result is known as a triangular number. Hence, numbers like 1, 1 + 2 = 3, 1 + 2 + 3 = 6 and so on, are triangular numbers. This question asks us to consider the possibility of double counting a number. So instead of 1 + 2 + 3 + 4, if you were to do 1 + 2 + 3 + 3 + 4, you would realize that the number you would get would be 13, which would be more than 10 (your correct addition) and less than 15 ( the sum of 1 to 5) , which is the next triangular number: And the double counted value could be achieved by spotting 10 as, the immediately lower value – and the difference between 10 and 13 would give you the double counted number.

To use this logic in the given question, we should realize that we just need to find the triangular number below 1000. A lot of engineers among you may be thinking about n x (n + 1) /2. However, there is an alternate way to solve this: 1 + 2 + 3 + 4 …. + 10 = 55. Hence, we can easily see that 11 + 12 + 13 + 14 + 15 + … + 20 would equal 155, and the sum of 21 to 30 would equal 255, and so on.

Thus, in trying to find the last triangular number below 1000, we can just do: 55 + 155 + 255 + 355 = 820 (which is the sum of the first 40 natural numbers) and since we have still not reached close to 1000, we start by adding more numbers as: 820 + 41 + 42 + 43 + 44 = 990 and the difference between 990 and 1000 is 10 which is the required answer.

For this question, we just need to carry the learning from the previous question forward. When we miss a number, we actually get a total, which is lower than the correct total. Hence, if we want to find the number missed, all we need to do is to find the first triangular number greater than 575. This can be got simply by
55 + 155 + 255 + 31 + 32 + 33 + 34 = 595, so the number missed has to be 20.

In this question, all you would need to notice is that the series, ABBCCCDDDDEEEEEF… ‘A’ ends after the first term; ‘B’ ends after the second term; ‘C’ ends after the sixth term, and so on. What we are looking at is how many numbers need to be added before we get to a number just below 288. So 55 + 155 + 21 + 22 + 23 gives us 276, which pretty much means that the 24th alphabet i.e. ‘X’ would be running in this series when we reach the 288 the term.

It is evident from the three questions and their solutions that careful dissections of the first question led to faster solutions to problems two and three. This goes on to highlight the importance of developing intelligence alongside your preparation.

If you have developed the ability to not only come up with solutions to complicated problems, but also have a mind map of how you arrived at it, it means you are well on the path of developing your cerebral skills. This will not only help you excel in management entrance exams, but in life in general.