In answering questions for the quantitative section of the GRE, it is paramount to be able to use smart approaches to avoid brute force calculations and to use as little time as possible that can be spared for other questions. The use of calculators, while not forbidden, should be limited to the bare essentials, i.e. almost never. Here is an interesting example to explain my point: Which of the following is larger: A) The sum of all odd integer numbers from 1 to 199 B) The sum of all even integer numbers from 2 to 198 ? Summing the numbers would indeed provide an answer, but the candidate would burn a lot of time in the process and get mentally tired as well. Let's look at a few numbers in these two series: 1, 3, 5, 7, 9, 11, ....... , 197, 199 2, 4, 6, 8, 10, 12, ....... , 198 The first row has (199+1)/2 = 100 numbers, the second row has 99. In the second row each number is one unit bigger than the one in the same position in... read more

If you are like me and share an interest in your family tree, you might have noticed that family trees get larger and larger with every generation. A legitimate question is how big can they really get. The answer is that they can get really big. Here is a simple example. Let's compute how many ancestors each of us has had over a period of 1,000 years. I will assume that every generation reproduces once every 30 years. Task 1. How many generations are there in a period of 1,000 years if, as assumed, each generation reproduces once every 30 years? Answer. 1000/30 = 33.33 which we round to 33 generations. Task 2. How many ancestors are there in 33 generations then? Answer. The number of our ancestors doubles every generation. In fact, we have 2 parents, 4 grandparents, 8 great-grand-parents and so on. Therefore, one should recognize the pattern, i.e., the number of ancestors goes up in powers of 2. Specifically, going back 33 generations, the number or our... read more

Many students I worked with tend to think that many notions they learn in mathematics are just that, notions and have little or no utility in real life situations. So, I like to give them problems, where they need first to derive their own equation or system of equations and then solve it. Here is a funny problem that never fails to surprise students: at first they feel lost for it sounds like a pseudo tongue-twister, but then they see the elegance of the solution emerge from within what sounds a hopeless situation. Problem: Mark is twice Bob's age and, when Bob will be Mark's age, together they will be 60 years old. What are Mark and Bob's ages right now? Solution: The "difficult" part is to write this problem in the language of a system of equations, actually a system of 2 equations. Let x be Mark's age and y Bob's age, respectively. Then we know that x=2y. For Bob to be Mark's age, we will need x-y years. At that point Bob will be x-year old and... read more

Often, students fail to appreciate the relevance of notions such as infinity and infinitesimal when they are first presented to them. And this is quite normal since both notions are challenging, in particular when they are combined. My favorite way to introduce them is to ask ancient Greeks for some help. More specifically I ask Greek philosopher Zeno of Elea (ca. 490-430 BC) to borrow his paradox of Achilles and the tortoise. The paradox goes like this: Achilles (whose speed among Greek heroes was legendary) is to enter a footrace with a tortoise. Achilles being a good sport allows the tortoise a 1 mile head start. We will assume that both Achilles and the tortoise move at a constant speed of "a" and "c", respectively. Now, Achilles has to run 1 mile to get to the point where the tortoise originally was. During this time the tortoise has run a much shorter distance, but will still be ahead of Achilles. Thus, it will take Achilles further time to... read more

During the Summer I occasionally receive inquiries from people who would like to learn some Italian or improve their conversational skills ahead of a trip/vacation to Italy. Basically, what they would like is to blend in a little and not look like the proverbial tourist. I tell them that some knowledge of "calcio" (soccer) goes a long way, but the truth is that there is nothing like coffee that can tell an Italian that someone is the infamous proverbial tourist unless some conventions are respected. Personally, I think that the list below represents the bare minimum of facts and unofficial rules one has to be aware: 1) DO NOT drink cappuccino, caffe' latte, latte macchiato or any milky form of coffee after a meal! Only in the morning! And, by the way, do not order a "latte" instead of "caffe' latte" or you will receive hot milk and strange looks! 2) DO NOT ask for mint frappuccino or the other concoctions that can be found at Starbuck's,... read more